With this solving method you have to look for a number that appears four times and which is located at the end of the diagonals of a rectangle. This number may only appear twice in a row (first example) or a column (second example). The two numbers on one of the diagonals will eventually be the solution.
First Example :
This is an example of a basic X-wing.
This X-wing has two identical pairs in cell D2 and H2 and two identical pairs in cell D6 and H6. Both pairs have the number 7 in common.
What is special about them? When you look at the example, then you see that the number 7 has to be in cell D2 or H2 and also in cell D6 or H6. If cell D2 is occupied by the number 7, then the cells H2 and D6 can not contain a 7, and cell H6 will also contain a number 7.
The same can be said if cell H2 would contain the number 7. In that case the number 7 has to be in cell D6. We see that there can only be a 7 in the cells D2 or D6 and in the cells H2 or H6.
Therefore all numbers 7 can be eliminated in the green columns.