X-Wings are fairly easy to spot, but a little harder to understand than some other techniques. Like others it relies on using positions of pencilmarks to infer enough to allow you to eliminate some other candidates.
X-Wings are when there are two lines, each having the same two positions for a number.
Take a look at this puzzle:
Once you've satisfied yourself that there aren't any easy methods you can apply to move forward, take a look at the candidate positions for 6, in rows 4 and 9.
The trick to understanding X-Wings is to imagine what would happen if you chose just one of those positions - what would it do to the others?
Imagine making the top left of those cells be the 6 - it would force the other candidate out on its row, and also force the candidate in the bottom left out too (the red arrows).
In turn, this would force the final cell to also be a 6 (the green arrow)
So a 6 in the top left cell, would force the bottom right cell to also be a 6:
By exactly the same logic, a 6 in the top right cell would force the bottom left cell to be a 6.
See how these two forcing lines form an X? That's how the technique got its name.
OK, I get the name, now what good does it do me?
If you think about it, whichever position 6 occupies in the top row, forces the other to occupy the opposite position in the bottom row.
Here's the clever bit - even though you don't know which row has the 6 at the left, and which row has the 6 at the right, you know for sure that both will be occupied
And because you know that the 6 will definitely be in both of those two column positions, you can look up and down those columns, and remove any other candidates!
We can't remove any 6s from the left hand column this time, but there are two we can remove from the right hand column, and one of those leaves an 8 as a single candidate!
What is new about this technique is that knowledge about two (similar) rows, lets you make removals from columns. Of course, it works the other way too, if you can spot similar columns. You'll often spot X-Wings - they are quite common, but they won't always lead to you being able to remove candidates.
The trick to spotting X-Wings is to look for the rectangles of possible candidates. If you find four candidates on the corners of a rectangle, check to see if they are an X-Wing for both rows and columns - that might save you some extra time!
Some more examples
X-Wing in rows for 8.
X-Wing in rows for 9.
X-Wing in columns for 4.