The software works by maintaining a list of the possible numbers that can go in each square, and gradually eliminating them as you go. Each time you click 'Next step' a chain of events take place:
Eliminate possibles based on numbers already placed. So if there's a 4 in the top left corner, remove 4s from the rest of the top row and the left column.
Eliminate possibles based on chevrons. For example, in a 5x5 puzzled, if square A > square B, then A cannot contain 1, and B cannot contain 5 (since 0 and 6 are not allowed).
Then there is a sequence of algorithms. To understand these better, load the appropriately named puzzle from the list, and click 'Next step' a few times until the algorithm is displayed.
Look for 'naked singles'. If there is only one possible, then of course that must be the number for that square.
Look for 'hidden singles'. If a possible number occurs only once per row or column, that must be the number we want.
Look for 'naked pairs'. If the same pair of possibles occurs twice in a row or column, with nothing else in those two squares, those two numbers can be deleted from other squares in that row or column. In the example, the 'green squares' must contain 4 and 5, or 5 and 4. Either way 4 and 5 can't go anywhere else in that column.
Look for 'hidden pairs'. If the same pair of possibles occurs twice in a row or column, but nowhere else in that row or column, any other numbers can be deleted from those squares. In the example, the two coloured squares must contain 4 and 5 or 5 and 4. The other posibles can go.
Look for 'X-wings'. If a number occurs in the same two positions in two rows, and nowhere else in those rows, that number can be deleted from the two intersecting columns (similarly for columns and rows). In the example, looking at columns 1 and 6, you can see that the 6s must occur top left and bottom right, or top right and bottom left. Either way the other 6s in those rows can go.
There are other algorithms that I haven't yet implemented. Do contact me if you have any comments or suggestions about this site.