(Notes on interactive Java applet: click to change the colour of a square; press 'u' to undo, or 'c' to clear the grid and start from scratch).
You have to fill a grid with black or white cells, so that: there is one contiguous region of each colour; no 2x2 region is filled with a single colour; for each numbered cell, the number indicates how many squares of continuous colour there are along the row and the column starting at the numbered cell.
Taking that a little slower:
Here's a small example of a completed 5x5 grid obeying these rules.
For example, if you are given a white square containing the number '2', then that square and exactly one of its immediate neighbours is white. All other immediate neighbours of the square (excluding diagonals) must be black, and the next square beyond the white neighbour must be black.
Here is an example of a completed grid:
(Stop reading now if you want to try to solve the puzzle yourself.)
The white '4' in the top left corner indicates that there are a total of four white squares starting from that corner and going out to the right or down. In this case, there are two possibilities for the 4 white squares: the whole top row and one square below the '4', or the whole left column and one square to the right of the '4'. Either way, the squares to the right and below the 4 must be white:
Next, look at the square in the middle on the right. If it's black, then either the 2x2 square in the bottom right is all black, or there's an isolated island of white in the bottom right of the grid - either of which is bad - so we fill the square in white.
There are a few rules-of-thumb that can be deduced from the basic rules after a bit of playing.
You should be able to figure these out for yourself: don't read them until you get completely stuck.
If you are completely stuck, you might find that these help you: