I'll have to give you the abbreviated answer, as I am about to leave work.
Typika is the shortest, a sort of "bare bones" prayer. It involves a couple of litanies, the Lord's prayer, epistle and gospel reading, a short supplication, and a Theotokion. You can insert other various prayers, though, as the situation allows.
Akathist is directed toward a particular saint, such as the Theotokos or one's patron saint. Usually these are done on festal days.
Vespers is done only at night, ideally at sundown.
Compline can be pretty lengthy, requiring a couple of hours--though I've seen them as short as 30 minutes. This is the one I prefer when we get together with friends to pray.
Reader's service is usually done as a replacement for Divine Liturgy. It involves reading the hours and the Typika. My parish does this on Sunday mornings when my priest is out of town and we cannot find a replacement (that's happening less often now).
Hope that helps. Others will probably be able to give you a more in-depth answer.