What disqualifies an expression from being linear? I'm taking some advanced math classes at my high-school, and I have some questions.
A helpful answer would include a definition. Examples of a concept or definition given (no matter how simple; please ELI5). 
This question I have historically phrased: "What makes a linear equation linear?"
I get that an equation that graphs out to be a line is linear, but it seems there is more to the meaning of "linear."
Additionally, why is linear algebra called linear algebra? What's linear about it?
 A: Perhaps the core idea of a function $f$ being linear* is the idea that if you change the input to $f$ in some specified way, it will affect the output the correspondingly. That is to say, in one dimension, a function $f(x)$ is linear if we can always say that $f(x+1)$ is $m$ bigger than $f(x)$ for some $m$ - that is, moving the input one value to the right corresponds to moving the output $m$ upwards. It is apparent that any function satisfying this is of the form
$$f(x)=mx+b$$
and that the description of "a move of $1$ to the left is a move $m$ up" corresponds to a geometric notion of a line. The convenience of this is that we can easily get from there to determining that, to increase $f$ by $1$, we must increase $x$ by $\frac{1}m$.
We can contrast this to functions like
$$f(x)=x^2$$
which do not behave so neatly - if we start with an input of $0$, then increasing it by $1$ yields a value $1$ greater - but increasing it by one again increases the value by $3$. This makes most operations with such a function harder than with linear functions - we can't so easily predict long term behavior based on a small sample, since the function might curve sharply or do other unexpected things.
In multiple dimensions, we end up finding functions of the form
$$f(x,y)=ax+by+c$$
where, for instance, increasing $x$ by 1 and $y$ by 1 increases $f$ by $a+b$ or increasing $y$ by $2$ increases $f$ by $2b$ - that is, if we pick any direction to move the input vector $(x,y)$, we still have the same "niceness" condition as before - that the change in $f$ will be constant, no matter what value we start at. We can, as before, easily predict how far we have to go in a given direction in order to elicit a given change in $f$ We could, more generally, formalize this notion as:

A linear function in one dimension is one of the form $mx+b$ - i.e. one whose graph is a line. In more dimensions, it is one which appears linear if viewed traveling on a single line; i.e. its restriction to a line in its domain is linear.

and it happens that these are exactly the functions of the form
$$f(x,y,\ldots,z)=ax+by+\ldots + cz + d$$
which tend to be very easy to work with.
Now, as for what this has to do with linear algebra. At an abstract level, linear algebra studies the concept of vector space - which, essentially, is a notion in which we can define two notations:


*

*We can add vectors together in a sensible way.

*We can scale vectors by a constant factor.
Which, essentially, is our way of defining a Euclidean (intuitively, "flat") space. A linear function, in this context, is one which preserves the above operations - so adding a given vector has the same effect regardless of basepoint and adding a multiple of a vector scales the effect of adding the vector. In particular, a linear function is defined as a function $f$ satisfying:
$$f(x+y)=f(x)+f(y)$$
$$f(\alpha x)=\alpha f(x).$$
It is worth noting that this notion of "linear" is a little stronger than the one used in the paragraphs before (which would be called "affine" functions) - the axioms imply that $f$ preserves the zero vector - i.e. $f(0)=0$, which is not true of the above. So a linear function actually starts to look like $f(x)=mx$ under this definition.
At some level, linear maps are the primary object of study in linear algebra - they define precisely what structure matters and what does not. In some sense, a linear map $f$ is one such that, if a set is flat, so is its image under $f$. This may all seem somewhat abstract, but it turns out that, in finitely many dimensions, matrices exactly correspond to linear maps - multiplying a matrix by a vector represents applying a linear function to that vector. Multiplying two matrices represents composing two linear functions. And, at the end of the day, for your intuition about the word "linear": Applying a matrix to a line (represented as a set of vectors) yields another line.
A: The idea of linearity is that the structure is conserved with the addition, and the multiplication by a scalar
About linear algebra :
If you have two vector spaces $E,F$ over a field $\mathbb K$, a function $f:E\to F$ is linear if $\forall x,y \in E, \forall a,b \in\mathbb K$
$$f(ax+by) = af(x)+bf(y)$$
Linear algebra is the study of such functions
About linear equation :
An equation is said to be linear if $\forall x,y \in E$ solutions and $a,b\in \mathbb K$, $ax+by$ is also a solution
A: There is a small cultural gap here: mathematicians don't usually talk much about "linear equations". However, if you pressed me to give you a definition, I would say that a linear equation is an equation of the form $f(x_1,x_2,\dots, x_n)=c$, where $f$ is a linear function.

There is also a large cultural gap here.
As Meelo mentions, there are two conflicting notions of a "linear function". Some look like
$$f(x_1,x_2,\dots, x_n) = a_1x_1+a_2x_2+\cdots+a_nx_n + b$$
and some look like this but without the $b$. This is simply an unfortunate choice that someone made at some point regarding the second definition: the first function has a graph which looks like a generic line (when $n=1$; vertical lines not included), and the second function is what most mathematicians mean when they say a function is "linear".
Both answers explain this latter notion of linear: it is a bit stronger (i.e. it is harder for a function to be this kind of linear than the other kind). Meelo also mentions in passing that mathematicians usually refer to the first kind of "linear" by the name affine. Unfortunately, this means that "an equation that graphs out to a line" is not given by a linear function (unless the line passes through the origin), but is a linear equation (as I defined it above). To make matters worse, a "linear equation", as I described above, looks a lot more affine than linear.
This may all seem hopelessly muddled and confusing to you; that's because it is. The two meanings of the word developed at different times and places, and the concepts became too useful in their context for the word to be replaced in either context. So we're stuck with it. This is really only a problem, though, at about your time in your math education. Give it a few more years, and you, too, will feel that faint annoyance anytime someone tries to tell you that $f(x)=x+3$ is "linear". For a mathematician, a function is not linear unless (in particular)
$$f(a+b)=f(a)+f(b),$$
and it is not true that $(a+b+3) = (a+3)+(b+3)$, since the right-hand side is $3$ bigger than the left-hand side.

However, you asked another question which I think is important: What's linear about linear algebra? The other two answers talk about this obliquely. The short answer is that linear algebra is the study of linear functions. But if you really want to know what's linear about linear algebra, it might help to see some examples of what's not linear about nonlinear algebra. This sort of algebra is just called algebra by mathematicians, although in university classes they usually append some adjective like "abstract" or "modern" to differentiate you from the remedial students :)
The answer to "what's nonlinear about algebra" is "approximately everything". Here is the simplest example of nonlinearity: you may have a nonlinear function; i.e. something like 
$$f(x,y)=-x^2+y.$$
This isn't a linear function. But it's still defined on a "linear space", a "flat" space. So we can ramp up the nonlinearity by saying that maybe our numbers don't come in a nice number line any more, so not like $\Bbb R$ but like $\Bbb C$.
But $\Bbb C$ is a lot like $\Bbb R^2$, which is flat, so maybe we could get even less linear: going to $\Bbb Z$ would work: the numbers are back to being ordered, but they're now too "far apart" to be linear.
Okay, but this is still pretty linear, because we still can multiply a number by another one to "move it a long the line", so let's remove multiplication: there's only addition. It's harder to explain a natural setting for this, although they show up all over the place in higher math. 
So now even the definition of a linear function that we gave before doesn't make sense, because we need multiplication: $a_1x_1+\cdots$. 
But wait, we can make the space even less linear: who says that $a+b=b+a$? This is called commutativity, and there are very natural ways to define "addition" where this is no longer true. This "addition" isn't anything like the addition we're used to, of course. We still call it addition because it's a rule for taking two numbers and getting out a new number, but the rule is less linear than ordinary addition.
Now we could still define linearity to be something like $x_1+x_2+\cdots+x_n$ (note there are no multiplications happening here, so we're safe), but does that mean that $x_2+x_1$ isn't linear, just because the variables show up in the wrong order?

After removing all this structure, what we're left with is the notion of what's called a group, which is still more structured than a "set", because we at least have this notion of addition, however alien it may be. In an [abstract] algebra course, groups are usually the first object that you study. In our description above we spent a lot of time thinking about the numbers and the operations, but you could do a parallel "loss of structure" story for the linear functions, and what you'd be left with (if you start at the right place) is an object we call a homomorphism; these are the next objects that you study.
After learning about them for a long time you learn something about the intermediate stages of the construction above: rings and fields. And once you have a field, you are finally in the position to put the linearity "on top" of it.
So it seems like groups are very far removed from linearity. But in fact, linear spaces are just a very special type of group, and linear functions are just a very special type of homomorphism. From this vantage point, linear algebra is exactly what it sounds like: the section of [abstract] algebra that deals with linearity.
