What do people mean by "finite"? Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite".
(For instance, Doron Zeilberger, although he is fond of April Fool jokes, does not seem to be joking here, or here.)
For such arguments to take place at all, the participants must share some understanding of what the word "finite" means.
I have never taken part in any such argument, but if I were to have to do so, I would nervously have to admit that I do not know what this shared understanding is, and I would ask if the others would mind spelling out (if only for my benefit) what it is that they think they are arguing about.
Just to anticipate two possible lines of ensuing discussion (assuming that I wasn't just greeted by stunned or embarrassed silence):
If (assuming for the moment that there was no dispute as to what a "set" is) someone were to say that a set is finite if and only if it can be put into one-to-one correspondence with a set of the form $\{1, 2, \ldots, n\}$ for some natural number $n$, this would obviously be open to the objection that it takes for granted a common understanding of the existence of a unique set of natural numbers, known to all the participants in the discussion. It might perhaps also be objected to because it apparently implies performing a possibly non-terminating search for such a number $n.$ But even if that is not a problem, it surely cannot be the case that a person cannot even be said to know what the word "finite" means unless they already accept the existence of the (uniquely defined) infinite set of all natural numbers.
(Zeilberger, for one, would presumably be inclined to make some objection(s) along these lines.)
If, on the other hand, someone were to put forward Dedekind's definition that a set is finite if and only if it cannot be put into one-to-one correspondence with a proper subset of itself, this could be objected to because it seems to imply that you can only directly show a set to be finite by proving something about the collection of all mappings of that set into itself, whereas this can hardly be what anybody has in mind when they insist that all mathematical objects must be finite, therefore it cannot be the agreed-upon common understanding of what the word "finite" means.
(Tarski's definition that a set is finite if and only if every non-empty collection of its subsets contains a maximal element is open to a similar objection. So too is Staekel's definition that a finite set is one that can be doubly well-ordered.)
So, some other definition of "finite" would have to be agreed upon; but what might it be?
Of course, the participants, whatever their other disagreements, might silently agree upon silence as the only appropriate response to such a ridiculously naive and ignorant question; but what if it had to be discussed, say for the benefit of a child, whose naivety and ignorance could be excused?
 A: If two people are arguing over whether or not “everything” is “finite”, then I'd say the difference between “cannot be put into one-to-one correspondence with a proper subset of itself” and “has a bijection to some set $\{1,\ldots,n\}$”, or any other definition of “finite”, is basically irrelevant. It's extremely unlikely they would suddenly agree if they meticulously chose a common definition, so why bother? (Consider an analogous situation with people arguing over the “existence” of “God”.) 
Remember, even if a formal mathematical definition was assigned a particular natural-language name (like “finite” or “smooth”), it's only because it is felt to capture some aspect of that concept (at least according to the namer). I'd say that people arguing over whether “everything” is “finite” have a disagreement regarding the natural-language concept of “finite” itself; they don't need to have agreed about which mathematical approximation to this concept they like the most.
A: Well. This is a tough cookie to answer properly.
The reason is simple, though. Finite is one of those words which has a mathematical definition, but also a natural language definition and those are so close that we might confuse the two.
This is similar to what does a set mean. Is a set some predefined notion, is it an element of a model of $\sf ZF$, or $\sf Z$, or $\sf NF$ or $\sf KP$, or maybe an object in the category $\bf Set$. Do every set has a power set? Do every definable subset of a set is a set?
These are notions which are fuzzy, specifically because they are taken as somewhat of primitive notions in mathematics.
But suppose that you have happened to agree upon some notion of "set", and let's agree to stipulate that it satisfied some naive set theory which is close in flavor to $\sf ZF(C)$.
Now you have several options:


*

*Claim that the natural numbers are not sets. They are urelements, or some atomic entities which satisfy the second-order axioms of $\sf PA$. Therefore the question what are the natural numbers is moot. And a set is finite if it can be mapped bijectively with a bounded set of natural numbers.

*Define the finite ordinals, claim that the class of finite ordinals is "definable" (either as a set, or as a proper class if you want to reject the axiom of infinity). Then prove that the finite ordinals satisfy $\sf PA$, so they are worthy of being called "The Natural Numbers", and we are reduced to the previous case.

*Use one of the many notions of finiteness which do not appeal to the natural numbers. These include, but not limited to, the following:


*

*Every self injection is a surjection.

*Every self surjection is a bijection.

*Every non-empty chain of subsets has a maximal element.

*Every non-empty collection of subsets has a maximal element.


Be forewarned, though, that apart of the last one, the axiom of choice is generally needed to prove that this is equivalent to the first suggested definition.
You may claim that the fact that there are definitions which are non-equivalent in the absence of the axiom of choice means that finiteness is not well-defined. And this is true. You can argue that you reject both the axiom of choice (and in fact, the axiom of countable choice), and the usual definition of finiteness. But you can also reject the axioms of induction in $\sf PA$ and claim that they are inconsistent, and you can reject the soundness of propositional calculus.
You can do all these things, but mathematics is a joint effort. If you are unwilling to agree on primitive notions like set, like finiteness, like natural number, then the problem lies in a deeper level than just this.
A: I don't see any interesting philosophical inquiry here. Your objections are purely argumentative.
If you are a finitist and believe that nothing infinite exists, then you have one very simple definition for finite: everything.
If you are not a finitist and accept the axiom of infinity, then you can use any suitable definition. With choice, (I think) they are all equivalent, and without choice you just need to be more specific as to what kind of finite you are talking about.
A: Your question strikes me as odd, because it asks "What do people mean by finite?" while simultaneously discussing two perfectly good definitions of the term.
A set $S$ is finite if there exists a bijection from $S$ to a section of the natural numbers.  Equivalently, $S$ is finite if every injection from $S$ to itself is a bijection.  These definitions are logically equivalent, so it makes no difference which one you use as a definition and which one you prove as a theorem.
You raise the objection that the natural numbers definition requires a common understanding of what the natural numbers are. That's a reasonable point, which is why mathematicians typically either define the natural numbers axiomatically or construct them as part of a larger axiomatic system. Historically, the natural numbers were defined axiomatically using the Peano axioms, but in the modern foundations of mathematics they can be constructed explicitly using the ZFC axioms for set theory. What this means is that the only common notions that are required to discuss mathematics precisely are the rules for symbolic logic.
Now, one can raise the objection that the axioms of ZFC may be inconsistent, in which case our entire discussion of mathematics is, from a formal point of view, entirely fruitless. Presumably this is what radical finitists like Zeilberger believe. Though we have no way to prove that the ZFC axioms are consistent, they have been working well so far, so the onus is on the finitists to demonstrate an inconsistency.  
A: Joel David Hamkins has argued that we do not have an absolute notion of the finite and has discussed this at length both in blogs and in publications (see e.g., the collection "a question for the oracle"). Namely the symbol $\mathbb N$ that we commonly use to denote an apparently clearly defined notion turns out to depend on the intended interpretation hypothesis. The stormy reception that question has received (14 upvotes, 12 downvotes) shows that the issue touches on some deep-seated beliefs among mathematicians.
A: Intuitively we may say that a set is finite if the process of listing its elements terminates. Formally  a set is finite if it is in a bijection with 1,2,...,n for some natural number n. 
A: Possibly, Kazimierz Kuratowski's definition is what you're looking for -- it's very straightforward.
Straightforward(!) but still a mouthful (more than other definitions), here is the Kuratowski's contribution:
Let $X$ be an arbitrary set. A family $\ L\subseteq 2^X\ $ is called a supfin family $\ \Leftarrow:\Rightarrow\ $ the following three conditions hold:
$$ (i)\quad\quad\quad \emptyset\in L $$
$$ (ii)\quad \forall_{x\in X}\quad \{x\}\in L $$
$$ (iii)\quad \forall_{A\ B\,\in\,L}
                     \quad A\cup B\in L: $$
Let $\ \Lambda(X)\ $ be the set of all supfin families $\ L\ $ in $\ X.\ $ Then we can let
$$ \mbox{Fin}(X)\ :=\ \bigcap\Lambda $$
be -- by definition -- the set of all $X$-finite sets.
DEFINITION (K.Kuratowski)$\ \ $ Set $\ X\ $ is finite $\ \Leftarrow:\Rightarrow\ X\ $ is an $\ X$-finite set.
This definition simply says that finite sets are obtained from nothing by adding single elements just a (hm!) finite number of times -- this definition is exactly what it is meant to be.
