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The game of Go is played on a square board composed of a net of vertical and horizontal lines - which produces a certain number of intersections. The size of the board is given by how many vertical lines (or horizontal lines - since the board is square) are used. So 2x2 board would have two vertical lines (and of course two horizontal lines) - which will make 4 intersections or, as they are called, four points. A 3x3 board would have 3 vertical lines (and three horizontal lines) - which produces 9 intersections, or points. These points are important because they are where you place your stones.
There are 3 standard board sizes: 9x9, 13x13, and 19x19.
Why these numbers? I honestly have no clue - except maybe for the 19x19 board. As you notice, all standard board sizes are odd-edged (so they have an odd number of vertical and horizontal lines) - this is to keep the board symmetrical - so you can have the point which is "the middle of the side" or "the middle of the board". What is the significance of such symmetry - I do not really know.
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The 9x9 board is the smallest of the standard boards. It is usually used for teaching beginners how to play. This is because of the fact that games on 9x9 board are faster (since the board is smaller) than on other boards and the complexity of the strategic issues is lower. Many of the concepts which are important (or even crucial) on larger boards do not play much of a role on a 9x9 board - for example a 3-space extension (discussed here) or the concept of moyo (discussed here).
But to say that a game on a 9x9 board is easy would be to fail to give it proper justice. Actually, the game on 9x9 board is fairly complex and demanding - although in a different way than on larger boards. On 9x9 board the main emphasis is on exact reading and calculations. It is a very tactical game, with lots of close-range fighting. Its a game in which the players are not distracted by the "big picture" and the whole-board complex strategies. You will find many strong players enjoying the games on 9x9 board.
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The 13x13 board is a step up in complexity from the 9x9 board. It is usually used for training players who already have a firm grip on the ideas which play a role on 9x9 board. As I said, 13x13 is more complex, and some of the intricate strategic concepts can be applied on it. The things which were sufficient to play a good game on a 9x9 board are no longer sufficient on a 13x13 - although they certainly play a large role too!
The 13x13 board is the "middle ground" which you step onto after you graduated from the 9x9 academy and before you're deemed strong enough to enter the 19x19 jungle.
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The 19x19 board is the board on which the major battles are fought! You might say that the experience of 9x9 and 13x13 was only necessary to lead you towards this one board, the ultimate challenge! It is here where all the complex strategic principles come to shine! It is here where your skill is tested to its limits!
Compared to 19x19 board, you can imagine the 13x13 board as the same game but with certain elements removed for simplicity's sake. The 9x9 would then be the same game still, but with even more elements removed. You use all elements of the game on 19x19.
Now - the question is "Why 19x19 and not, for example 21x21, since we need, for some reason, odd numbers? What is so special about the number 19?" - Well, the answer is simple - and you can find it here.
None, really. You might stumble upon a 7x7 now and then - but not too often. I have seen a couple of 11x11 boards in my time - but not often either. The three standard sizes (9x9, 13x13, and 19x19) are all there is used out there, and they account for 99.999999999% of all the games and 99.9% of the Go-related considerations. These numbers are, of course, only approximations and not the results of some exhaustive studies.
The reason why there seems to be a discrepancy between the two above numbers (for actual games and for Go-related considerations) is that there is a group of people seriously investigating the various game possibilities on boards like 2x2 or 3x3 or 4x1 - and all kings of other weird sizes. They do it primarily to investigate what effect the various rule-sets have on the game. They also seem to subscribe to a theory that the principles they can come up with by investigating such small boards would give them some insights into the strategies on 19x19. Maybe they are on the right track, or maybe they are nuts - I do not know. The fact that some quite serious and knowledgeable people work on this is certainly an indication that there must be something in it - the time will tell. For now - it just seems to be an attempt to "westernize" the Go knowledge by applying some of the methods we have used to build up our science and technology - namely to start with a small problem, devise a set of fundamental axioms, and build up on these axioms logically until you find out what makes the universe tick! Or or what makes the universe ticked, hehe.... Anyways - they still seem to be looking for these fundamental axioms.... but who knows where this path may lead - so it certainly should net be discarded too easy.