Eliminate Pairs, Triples and Quads
This is one of the simple solution methods which you will use regularly to solve the puzzle.
Example: after solving apart of the puzzle, you have gotten the following situation:
In our example similar pairs appear in cells D6 and F6. Both cells contain the numbers (1,3). This means that if number 1 is the solution for cell H6 then number 3 will be the solution for cell F6 or the other way around.
Since each number can only appear once in every row, the numbers 1 and 3 can not appear in the other cells, so they can be eliminated.
Cell H6 contains the numbers (1,9). Number 1 can be eliminated, leaving number 9 as the solution.
The numbers (1,3) can only appear once in the 3x3 box, which results in the elimination of number 1 in cell D5 and F5.
If you scan the column, you will find similar pairs (4,9) in the cells G2 and G3. This will result in the elimination of the numbers 4 and 9 from the cells I1, H2 and H4.
If you find a box containing two similar pairs then these numbers can be eliminated from the other cell in the same box.