To expedite something is to make it go faster or speed up the process. Expediting that package might get it to Alaska by tomorrow, instead of next Tuesday.
If someone says, "Let me expedite the process," that's probably a good thing: they're offering to speed things up. If you've ever waited in line a long time, then you must have wished someone could expedite things. A driver can expedite his commute by going in the faster car-pool lane. Whenever there's a lot of paperwork for something, it's nice to know someone who can expedite the process. When you see the word expedite, think "Make faster!"