To speed up the process or progress of.
Example:
The sales department was looking for ways to expedite the shipping and billing of incoming orders.
Expedite comes from the Latin verb as expedient, but expedite usually indicates only speed or efficiency and doesn't involve moral issues at all. Many people concerned about health-care issues, for example, have campaigned to get the FDA to expedite its approval of new drugs. And new kinds of educational software are expected to expedite the learning process.