Recount has two distinct meanings:

1. To tell a story or describe events that happened. When you recount your day to a parent at dinner, you're describing what happened in order: first this, then that, then something else. A witness might recount what she saw at an accident. Historical accounts recount battles, discoveries, or important moments from the past. The word suggests a careful, detailed telling rather than a quick summary.
2. To count something again to make sure the first count was accurate. After a very close election, officials might recount the votes to verify who actually won. If you're counting a large jar of pennies and get 247, then count again and get 253, you'd better do a recount to figure out the real number. A recount (as a noun) is that second counting process, often done when the results matter and people want to be absolutely certain the count is correct.

Both meanings share the idea of going back over something: either retelling events or counting numbers again to get them right.