So in mathematics we measure the cardinality of a set with bijective functions/maps or one to one correspondences.

For example suppose you know that there are 100 seats in some movie theatre. When the movie starts, suppose it is a hit movie and fills up. In other words, there is a person for every seat in the theatre. Without counting the number of people, we can deduce that there are 100 people in the theatre. This is an example of a one to one correspondence (also known as a bijective function or map) between people and seats in the theatre, i.e. the cardinality of the people is the same as the cardinality of seats because for every seat there is one person sitting in it, and for every person there is one seat that they are sitting on.

There are two types of sets, countable and uncountable sets. Countable sets can either be finite or infinite, but uncountable sets are always infinite just a 'larger' infinite.

More precisely, A set X is finite if there is a bijection between the set X and the finite whole numbers, N_n={1,2,3,...,n}. If X is not finite, then X is infinite (they mean the same thing). Now concerning infinite sets, there are two types, countable and uncountable (here is the difference you seek). An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N. Note: Finite sets are also countable.

I think examples will be helpful here:

The set A={1,2,3,4,5} is finite and countable.

The set of integers is considered *infinite *and countable.

The set of real numbers (rational numbers and irrational numbers) is infinite and uncountable.

You can, informally, think of a countable set as a set where you are able to potentially list all of the elements of the set, and think of an uncountable set as saying there is no list that contains all the elements of the set. Naively, we can see that the real numbers are uncountable, because between any two real numbers there is another real number. Whereas there is no integer between the numbers 1 and 2.

Hope this helps!

Links for further reading:

http://gowers.wordpress.com/2011/11/28/a-short-post-on-countability-and-uncountability/

http://en.wikipedia.org/wiki/Cardinality

Here is some elaboration on the comments.

What Cantor showed was more precisely that if $X$ is any set, then the powerset of $X$ (that is, the set consisting of all subsets of $X$, written $\mathcal{P}(X)$) is strictly larger than $X$, even if $X$is infinitely large.

To make sense of this mathematically, we need to define what it means for one set to be strictly larger than another. One way to do this is to instead define what it means for one set to be at most as large as another, and then negate that.

I will for simplicity use the definition that $X\leq Y$ if there exists a surjective map $f: Y\to X$ (usually it is defined using an injective map in the other direction, but at least assuming the axiom of choice, these definitions are equivalent, and it makes the following argument simpler).

Now, to show that $\mathcal{P}(X)$ is strictly larger than $X$, we just need to show that $\mathcal{P}(X)\leq X$ is false.

So how to show that there cannot be a surjective map from $X$ to $\mathcal{P}(X)$? Well, this is where Cantor came up with a nice "trick".

Let $f: X\to\mathcal{P}(X)$ be an map. We wish to find some element in $\mathcal{P}(X)$ which is not in the image of $f$. To do this, we define the subset $Y$ of $X$ as follows: $Y = \{x\in X | x\not\in f(x)\}$. This is certainly a subset of $X$, so it is an element in $\mathcal{P}(X)$, and I now claim that this element is not in the image of $f$.

To see this, assume for the sake of contradiction that $y\in X$ is given such that $f(y) = Y$. Now we know that we must have either $y\in Y$ or $y\not\in Y$. But if $y\in Y$ then by definition of $Y$, we have $y\not\in f(y)$ but since we assumed $f(y) = Y$ this is a contradiction.

On te other hand, if $y\not\in Y$ then again by the definition of $Y$, we must have $y\in f(y)$ (since otherwise, we would have $y\in Y$. But again, since we assumed $f(y) = Y$ this is a contradiction.

All in all, our conclusion is that there cannot be an element $y\in X$ such that $f(y) = Y$, so $f$ is not surjective. But since this was for an arbitrary function $f$ from $X$ to $\mathcal{P}(X)$, there can be no surjective function from $X$ to $\mathcal{P}(X)$.

The previous arguments do not, however, show that there are any infinite sets at all, but if we want to have a theory of mathematics that includes all the natural numbers for example, we need to assume that some infinite set exists. And once we have one infinite set, the above arguments show us that we can always find a strictly larger set than any given one (as long as we are allowed to take powersets, but it would be hard to work without this).

Finally, it should be noted that if you are not used to working with this concept of "size" of sets, then it might seem obvious that we can always get a larger set, by just adding an extra element. But it turns out that once you start looking at infinite sets, it takes some work to make them strictly larger, so for example, taking the cartesian product of two infinite sets does not result in a strictly larger set.

1. something that is finite is something that has ends or limits, it's opposite is perhaps a slightly more common word, infinite, meaning without limits or ends, everlasting etc.

2. probably, certianly most things do have a finite lifetime, but it's a rather philosphical statement so either view could probably be supported. e.g. if everything in the medium of time ends what about time itself?