¹Definition: An X-Wing deals with a single candidate number. The pattern forms when the candidate appears exactly twice in each of two parallel lanes, and the candidates line-up along their cross-lanes to form the corners of a rectangle. An illustration will make this much clearer:

²

³Look For: A lane in which a candidate number appears exactly twice. Then see if you can find a similar situation in a parallel lane. (i.e. the candidate number appearing exactly twice in the Lane) and where the numbers line up on the cross-lanes.

⁴Consequence: If you find the candidate number anywhere along the cross-lanes other than the X-Wing corners, you can erase those candidates from those cells. In our illustration, the 5's on C1, C8, and G4 can all be erased.

⁵Why it Works: When there are exactly two of a given candidate number in a Lane, we have what we will call a binary lane. A binary lane simply means that one of the two candidates in that Lane must be true and the other must be false (though we don't yet know which is which.)
With an X-Wing we have two parallel binary lanes with their cross-lanes lined-up. In effect this makes the cross-lanes binary lanes as well, since the original binary lanes have one true and one false end, and we can't have two true in the same cross-lane. So we know that the true ends will be diagonal from each other in the rectangle of the X-Wing. We can draw imaginary diagonal lines between the four corners: forming the X that gives this technique its name.
In our example this means that we know that a 5 will have to go in either C3 & G6, or in C6 & G3 (but in no other combination.) Because we know that a 5 will go in two of these corners of the X-Wing, and both combinations include columns C & G, we know that the 5 is accounted for in these columns, and so we can erase any 5's that occur in these columns other than on the X-Wing's corners.[We already know there are no 5's to erase on the rows, because that is how we found the X-Wing in the first place: there were only two 5's on the rows.]

⁶In our full board example from a real puzzle, looking down the first column we notice that the candidate 3 appears exactly twice.
Scanning the columns to the right, we find the same situation in the last column, and the 3's line up on the same rows: an X-Wing if there ever was one! Next, we find the3's on these rows that are not part of the X-Wing's corners, and we erase them.