Sudoku strategies:

Number-pairs:

If a group, row or column has only two occurences of a number-pair, then all other possibilities for those squares can be eliminated.

Variation: there may be more than 2 occurences of the number-pair but they can be eliminated if the number-pair is a chain.
Example:
{1, 7}, {6, 7, 9}, {1, 6, 7, 9}, {1, 7}, {1, 4, 7, 6}, {2, 3, 6, 7}, {3, 4, 6, 8, 9}, {2, 3, 4, 6, 8}, {5}
You can see that there are two cells that have the same two candidates 1 and 7. One of these cells must hold the 1, and the other cell must hold the 7, so 1 and 7 can be removed from the candidates for the other cells. This reduces the candidates to:
{1, 7}, {6, 9}, {6, 9}, {1, 7}, {4, 6}, {2, 3, 6}, {3, 4, 6, 8, 9}, {2, 3, 4, 6, 8}, {5}
The same is now true for the (6,9) number-pair.
By elimination we end up with:
{1, 7}, {6, 9}, {6, 9}, {1, 7}, {4}, {2, 3}, {3, 8}, {2, 3, 8}, {5}

Reducing options:

If a number exists only in one row (or column) of a group, then that number can be eliminated from the corresponding row (or column) in the adjacent groups. (Previously stated as: If the possibilities for a number in a group (2 or 3 possibilities) only exist on the same row (or column), then that number can be eliminated from that row (or column) in the adjacent groups.)

If a number occurs in only 2 rows (or columns) in a group and also in the same 2 rows (or columns) in an adjacent group, then that number can be eliminated from the equivalent cells in the third group.

Corrollary:
If a number cannot be in a particular row (or column) of a group, and also cannot be in the same row (or column) in a related group, then the number can only be in that row (or column) in the third group.

Cleaning up chains:

If a chain exists, then all additional possibilities in the chain can be discarded.
Chains of 2 numbers require 2 cells, chains of 3 numbers 3 cells and so on.
Example:
{4, 5, 6, 9}, {4, 9}, {5, 6, 9}, {2, 4}, {1, 2, 3, 4, 7}, {1, 2, 3, 7}, {2, 5, 6}, {1, 2, 7}, {8}

This group has a 3-number chain for the numbers (1, 3, 7): {1, 2, 3, 4, 7}, {1, 2, 3, 7}, {1, 2, 7}. Clearly the three numbers can only be in these three cells and no other numbers are possible. We can therefore simplify those cells to read: {1, 3, 7}, {1, 3, 7}, {1, 7}.

X-wing:

If a number only occurs twice in a particular row and the same number also only occurs twice in the same positions in a different row (or column), then that number can be eliminated from all other cells in those columns.
Example:

row 1: -*1 --3 6-* (where * indicates where a 9 might be placed, for example)
row 9: -*6 --2 4-*

Clearly in each column the 9 must be in one of the marked cells and so cannot be in any other cells in those columns.