Defining Material Conditional Defining the conditional statement "$\phi \implies \psi$" so as to capture what is meant by implication requires that we ignore the issue of causation.
From Keith Devlin's Introduction to Mathematical Thinking:

$\quad$The truth or falsity of a conditional will be defined entirely in terms of the truth or falsity of the antecedent and the consequent. That is to say, whether or not the conditional expression $\phi \implies \psi$ is true will depend entirely upon the truth or falsity of $\phi$ and $\psi$, taking no account of whether or not there is any meaningful connection between $\phi$ and $\psi$.
  $\quad$The reason this approach turns out to be a useful one, is that in all cases where there is a meaningful and genuine implication $\phi$ implies $\psi$, the conditional $\phi \implies \psi$ does accord with that implication.
  ...
  $$
\begin{array}{ccc}
\phi & \psi & \phi \implies \psi \\
\hline
T & T & T \\
T & F & F \\
F & T & ? \\
F & F & ? \\
\end{array}
$$
  ...
  $\quad$Once we get used to ignoring all questions of causality, the truth-values of the conditional seem straightforward enough when the antecedent is true...[b]ut what about when the antecedent is false?

Intuitively, we can see that $\phi$ implies $\psi$ means that it does not happen that $\phi$ is true and $\psi$ is false, as this captures the hypothetical aspect of implication, since we would like to consider an implication as true when the consequent does follow from the antecedent, even if the antecedent turns out to be false. Consider the following conditionals in each of which the antecedent is false:  
$\quad$If it is true that polygamy is legal in New York, then marriage laws in New York are not identical$\quad$with those in Nevada. (Here the consequent is true).
$\quad$If it is true that polygamy is legal in New York, then a man living in New York can legally have $\quad$three wives. (Here the consequent is false).  
Few would deny the truth of the preceding conditionals. From Introduction to Logic by P. Suppes:  

Although we have already admitted that the notion of connection or dependence being appealed to here is too vague to be a formal concept of logic, in choosing examples which will force upon us, within our truth-functional framework, a truth value for implications with false antecedents it is reasonable to pick an example...for which our intuitive feeling of dependence is strong rather than an example...for which it is weak.

And the reason why this is a reasonable thing to do is because there is a hypothetical aspect of implication that we would like to capture in our defining the conditional. Thus, we can agree that the truth table for $\phi \implies \psi$ and $\lnot (\phi \land \lnot \psi )$ should have the same truth-value for each particular combination of truth-values for $\phi$ and $\psi$.
$$
\begin{array}{cccc}
\phi & \psi & \lnot (\phi \land \lnot \psi) & \phi \implies \psi \\
\hline
T & T & T & T \\
T & F & F & F \\
F & T & T & T \\
F & F & T & T \\
\end{array}
$$
First question: Is this convincing and appropriate to use as an explanation for the definition of the conditional? Also please provide your own approach.  
Again from P. Suppes:

The truth functional demand...has no undesirable effects, since conditional sentences whose antecedents and consequents are unrelated and whose antecedents are false play no serious role in systematic arguments.   

Even though this is the case, I feel that it is important for 1) the conditional to be defined in all cases and 2) to provide an explanation for this definition. However, it is often said that the assignment of truth values for the case when $\phi$ is false is arbitrary, but that this produces a useful logic. This seems to imply that the conditional could have a different truth table and that we need not provide an explanation that appeals to our intuition of how implies should work. Second question: Is this true?  
Response to Derek Elkins:
At this point in my self education (I am looking at Modern Introductory Analysis by Dolciani, a precalculus text) I am looking for a solid explanation about implication and how we come to define the conditional. Most of your answer contains material that is currently out of my depth, and I am seeking an answer that will put to rest my doubts about my understanding of implication in propositional logic. In fact, I'm only even aware of first-order logic and propositional logic not as theories, but more as a way to make communicating about mathematics precise. Given your extensive knowledge and my lack thereof, and bearing in mind that I am self-studying mathematics at the precalculus level, can you provide me with an explanation that you found satisfying when you were at this level? Also, can you please share with me online resources, or books, that helped you in your education? Thank you. 
 A: A definition doesn't have to be convincing or appropriate; you can define things however you want.  Of course, it is better to have a convincing argument on why a definition is appropriate and corresponds to what is being modeled.  You can definitely have good and bad definitions.  If we did change the result for $\phi$ false, you'd get logical equivalence.
If you define your logical connectives in terms of truth tables, then you don't have the choice of not defining implication for all inputs.  However, truth tables are a semantics for propositional logic and just because a proposition is satisfiable doesn't mean that it is (syntactically) entailed.  Of course, for the standard rules of propositional logic we have soundness and completeness theorems that state exactly that these two notions coincide.  If we change the rules of logic, though, we can formulate a logic where $\phi \implies \psi$ is not provable (i.e. not syntactically entailed) when $\phi$ is false, for example relevance logic.  This is roughly what "not defining implication for false antecedents" would mean at the syntactic level.
To directly answer your first question: within the context of standard propositional logic the answer is unambiguously "yes".  Soundness requires that two logically equivalent formulas have the same truth table.
However, this just pushes the problem back to "should those two formulas be logically equivalent?"  To which many people, including me, say "no, not in general".  This is (one of) the difference(s) between classical and non-classical (and in particular constructive) logic.
Ironically, that logical equivalence that you present as a possible justification for $\phi \implies \psi$ is true when $\phi$ is false and $\psi$ is true, doesn't hold in (non-classical) constructive logics, but the statement itself [EDIT: $\phi \implies \psi$] does hold in most constructive logics (particularly in ones that are not substructural [note that relevance logic is a non-classical, substructural logic.])  Since classical logic is just a special case of a constructive logic, this means that there is a better, more fundamental reason why implication should behave the way it does.
The BHK interpretation provides one approach to this.  If we think of propositions as true when we have a proof object for them, then logical connectives like implication are operations on these proof objects.  In particular, in the BHK interpretation implication is interpreted as a function that takes proof objects of $\phi$ and produces proof objects of $\psi$.  With this interpretation, the fact that implication can be true when its antecedent is false is merely the fact that functions can ignore some of their arguments.  Indeed the analog of the BHK interpretation for relevance logic is exactly a lambda calculus where functions are not allowed to ignore their arguments.
The Curry-Howard correspondence, by grounding logic in computation, provides, for me, the real explanation and justification for logical notions.  Admittedly, this is a fairly radical view.
A: 
This seems to imply that the conditional could have a different truth table and that we need not provide an explanation that appeals to our intuition of how implies should work. Second question: Is this true?

The first issue is that treating $\Rightarrow$ as a truth function already removes much of our intuition about implications, because now the truth of the implication can only depend on the truth or falsity of the sentences involved, not on their actual meaning.
In any case, if we accept that $T \Rightarrow T$ is true and $T \Rightarrow F$ is false, there are four possible truth tables for $\Rightarrow$, which I will list with ugly formatting. 
$$
1.\quad \begin{matrix}
 & T & F \\
T & T & F \\
F & T & T 
\end{matrix}\\
2. \quad
\begin{matrix}
 & T & F \\
T & T & F \\
F & T & F 
\end{matrix} \\
3. \quad
\begin{matrix}
 & T & F \\
T & T & F \\
F & F & T 
\end{matrix} \\
4. \quad
\begin{matrix}
 & T & F \\
T & T & F \\
F & F & F 
\end{matrix} 
$$


*

*Option 1 is the usual definition of $P \Rightarrow Q$  

*Option 2 would make $P \Rightarrow Q$ be the same as just $Q$

*Option 3 would make $P \Rightarrow Q$ be the same as $P \Leftrightarrow Q$

*Option 4 would make $P \Rightarrow Q$ be the same as $(P \text{ and } Q)$. 


As you can see, there are already names for the other three options, and none of them acts like our intuition says that implication should act. The truth table we use may not seem ideal, but it turns out to be better than the alternatives.
A: 
" it is often said that the assignment of truth values for the case when  is false is arbitrary, but that this produces a useful logic. This seems to imply that the conditional could have a different truth table and that we need not provide an explanation that appeals to our intuition of how implies should work. Second question: Is this true?"


"Although not explicitly stated in my post, I too, agree that the truth table is better than the alternatives. In fact, I think it captures a whole lot about genuine implication. However, I am interested in being confident in my explanation of why we should consider a conditional statement as true whenever the antecedent is false. What was the thinking behind this? Was it because it captures the hypothetical aspect of implication, or because it is not the case that the antecedent is true and the antecedent false?"

The question seems to be about the philosophical underpinning of either the "Principle of Explosion" ("from falsehood, anything follows") and/or the notion of "vacuous truth".
I don't have definitive answers to any of this, but just some comments that are too long to be a comment.
(1) If you accept that propositions must either be true or false (a.k.a. the Law of the Excluded Middle), i.e. there are no propositions that are "neither true nor false", then the accepted answer to this question shows how the definition of material conditional is the only one that works.
(2) There are many ways in which it has been noted that the "principle of explosion" or "vacuous truth" following from the definition of material conditional in classical logic do not match our intuition, i.e. you are not alone in finding this unusual. See the Wikipedia article "Paradoxes of material implication".
(3) Given (1), we are left with two options: accept that this is the best we can do, or try to weaken the assumptions we made in (1) to look for hopefully a "better" result. Which option you choose ultimately comes down to your personal feelings, namely whether

*

*you are more uncomfortable with abandoning the Law of the Excluded Middle than you are with retaining the [Principle of Explosion]/[vacuous truth]/[paradoxes of material implication], or

*you are more uncomfortable with those paradoxes than with abandoning the Law of the Excluded Middle.

A quote of von Neumann referenced in a comment to a related quesion is relevant here: "in mathematics you don't understand things, you just get used to them". (I also agree with the commenter that "I don't think that's always good advice, this might be one of the times when it is".)
Speaking personally, I am familiar with my discomfort with vacuous truth, whereas I am not familiar with all of the intricacies of logical systems where thte Law of the Excluded Middle is not valid. So I am more uncomfortable abandoning the Law of the Excluded Middle, even in the absence of what I find to be any compelling philosophical underpinning or justification for the notion of vacuous truth.
Anyway this is one reason why the first answer by Derek to this question is complicated - even if the material conditional does not entirely "make sense" in classical logic, you either have to accept it how it is, or you have to try out fundamentally distinct formal logical systems besides the "classical logic" taught in the vast majority of textbooks.
(4) Even if you decide to give up the Law of the Excluded Middle, and the accompanying constraints of a $2 \times 2$ propositional truth table, you might still want to accept the Principle of Explosion, because it still makes things simpler.
Quoting from an answer to a related question:

To give an explicit example, we can easily prove in [with principle of explosion but without law of excluded middle] that if a natural number $n$ is not prime, then there are natural numbers $a,b$ such that $1 < a$, $b < n$ and $n = ab$. But this won't work [without both the principle of explosion and the law of the excluded middle].

Heuristically speaking, so-called "intuitionistic logic" is a formal deductive system similar to the "classical" logic except that it rejects the Law of Excluded Middle.
Heuristically speaking, so-called "minimal logic" is similar to "intuitionistic logic" except that it additionally also rejects the Principle of Explosion, and thus does not have a notion of "vacuous truth", cf. this answer to a related question.
Wikipedia also gives the following philosophical argument for the Principle of Explosion, which I personally find at least somewhat compelling, even if it doesn't make my uneasiness with the notion of "vacuous truth" completely go away.

The ... value of the principle of explosion is that for any logical system where this principle holds, any derived theory which proves [something false] is worthless because all its statements would become theorems, making it impossible to distinguish truth from falsehood.

(5) There appear to be other "Paradoxes of material conditional" besides those stemming from the notion of "vacuous truth" / the Principle of Explosion that the above doesn't address (and with which I am less familiar).
According to Wikipedia, so-called "connexive logic" and/or "relevance logic" are other formal deductive systems that attempt to address these or other "paradoxes of material implication". I won't pretend to know or understand anything else about them. The first answer seems to at least mention them.
(6) Techically the above actually somewhat conflates the "Law of Excluded Middle" with "Principle of Bivalence" = "Law of Non-contradiction", cf. this answer to a related question. I.e. the derivation of the truth table in the accepted answer only requires the Principle of Bivalence, from which the Principle of Explosion follows, cf. this comment for a related question. However, the fact that a "truth table" would only have two rows and two columns does follow from the Law of Excluded Middle. So I think it still makes sense to identify the Law of Excluded Middle as something that needs to be abandoned/relaxed if we want to explore possibilities for a "better material conditional", cf. this other comment for a related question.
