regular - regular expression, regular grammar, accepted by a DFA, NFA or NFA-lambda
context free - context free grammar, PDA (pushdown automaton)
context sensitive - not covered in detail, rules like Ab = Abb
recursive - Turing machine that always halts, uses final states
recursively enumerable - Turing machine that halts to accept, (no final states) but otherwise either does not halt, or halts abnormally
The complement of a recursive language is recursive. - If it's recursive, it's accepted by a Turing machine that always halts. If you flip the accepting/non-accepting states to get the complement, the machine still always halts, so the complement is recursive.