I understand why you would refer to mathematics which concerns itself with all of the numbers on the number line as 'continuous' but why would you refer to countable or finite mathematics as 'discrete'? Like it's the mathematics no one talks about? Where does this labeling come from?
[Math] Why is it called ‘discrete’ mathematics
discrete mathematicsmath-historyterminology
Related Solutions
The difference between countable and uncountable sets is well formalized and there is never any doubt. These are two different "sizes of infinity". You can read this page for more information on why countable is not the same as uncountable: http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
But I think one intuition which is really helpful, and also linking this with computer science, is the fact that a countable set is a set whose elements are finitely describable. For instance each integer can be written on a piece of paper, so the set of integers is countable. This makes integers manageable by computers: since you can completely describe an integer in a finite way, you can always pass it to a computer, as a finite sequence of $0$'s and $1$'s. This is also why reals are not countable: you might need to write down all the decimals, that is an infinite sequence. This makes "continuous mathematics" not well-suited for automatic treatment by computers.
Of course this is very schematic and can be further detailed, but this intuition is very important. It is possible to formally prove: "every element of $E$ contains a finite amount of information $\implies$ $E$ is countable".
Following this intuition, rationals are countable, because a rational $r$ can be given by two integers $a,b$ with $r=a/b$. This does not prevent rationals to be an important tool of analysis, because reals can be approached by rationals arbitrarily close (we say $\mathbb Q$ is $dense$ in $\mathbb R$). But most computer algorithms dealing with "arbitrary" numbers actually deal with only rational numbers.
As for the classification of maths into "discrete" and "continuous", the frontiers are really not well-defined, and everything interacts with everything else, so it is almost impossible to give a sound definition. A big part of it is subjective. At best, you have a "flavour" in some fields that is mostly discrete (like graph theory) or continuous (analysis), but in both cases, you might need also to consider the other side in order to get a good understanding (like using probability theory in graph theory).
There are different ways to think about the size of a set. In the case of the real numbers, and specifically intervals, we can talk about their length (and generally, their Lebesgue measure in the case of measurable sets).
If you think about the real numbers as a model of time or space, then the distance between you and the screen through which you are reading this is a finite interval. But in this model, based on the real numbers, it is an uncountable interval, not a finite set.
One thing to remember about terminology, is that it should highlight to the reader or listener something about a certain relevant property. In the case of intervals, we already know they all have the same cardinality (in the case of non-degenerate intervals). So we can use "finite" or "infinite" to talk about their length (and formally, their measure). Thus setting the importance on that aspect, rather than their cardinality.
Best Answer
As the online dictionary at m-w.com says, "discrete" means anything constituting a separate entity or anything consisting of distinct or unconnected elements.