There are a number of strategies for solving Sudoku, ranging from the very simple to ones that might make your brain overheat a bit. The simpler puzzles can often be solved using only the simpler problem solving techniques. For the hardest puzzles you'll have to use them all. We'll cover them one by one - first of all, "line drawing".
(I should mention here that there don't seem to be standard names for the different techniques, but we have to call them something - so "line drawing" it is.)
Line Drawing: this is the most straightforward technique, and the easiest one to use if you're solving a sudoku on the train on your way to work. It doesn't involve lots of counting and marking, and it can often solve quite a bit of the puzzle.
Let's look at this board and think about where we could place a 5 in the upper left box.
The second row also already contains a 5, at position [9,2], so we can mark all of the second row in green as well.
Finally, we have 5 near the lower left hand corner, in position [2,7], so we can mark all of its column in green.
Now that we've marked these areas of the board we can have a look at the 3*3 box at the upper left, and note that of the 9 squares, we've marked 7 as being unable to contain a 5. Of the other two squares, one of them [1,3] already contains the number 1, which leaves us with just one square [3,3] which could contain the 5. Since every 3*3 box must contain a 5, this square must contain a 5 - and we've solved our first square of the puzzle.
Now a couple of pretty obvious things, but I'll say them anyway:
Let's look at this board and think about where we could place a 4 in the left hand column.
There's a 4 at position [4,2], so we know there will be no others 4s on this line, and can draw a line across it. There's another 4 at [7,5], so we know are no more 4s on this line either. Finally - and a little differently - there's a 4 at [3,7]. This last 4 is particularly useful as it doesn't just eliminate a single row - it also tells us there are no other 4s in the lower left hand box, thus eliminating three squares from the left hand column.
Let's colour in the squares and see what's left in the left hand column. When we eliminate the 8, the 3 and the 7 which are already filled in, we're left with only one square where the 4 could go - [1,6].
Next: counting techniques in Tutorial 2