A strip-cut shredder cuts the paper into strips between one-eighth and one-half inch thick; the smaller the strips, the more difficult it is to reassemble the document.
A cross-cut shredder lessens the volume of the shredded paper, so that the wastebasket doesn't need to be emptied as often.
Shredding sensitive information now may save you a lot of money if someone applies for a credit card or a bank account under your name...