Pushing and pulling is using a different ISO speed than what the film stock is rated for. If you have Tri-X 400 and shoot it as if it were 800 (or 1600, or anything higher and more sensitive), you're
pushing the film. You then develop it with some adjustments and the result are pictures with more contrast and punch, stark deep shadows and bigger grain size. A lot of the look associated with Tri-X is actually from it being pushed.
If you have the same Tri-X 400 and shoot it as if it were 200 (or 100, or anything lower and less sensitive), you're
pulling the film. Develop it with the appropriate adjustments and you get less overall contrast, but a lot more shadow detail and smoother tonality, and the grain takes on a different look.
Here's
a good article on these things :D