It is certainly true that mosquitoes need still water to breed, their larvae being aquatic. Depending on the species, this can be anything from a pond to a tiny puddle or even a thimble-full of water in a bromeliad. The main requirement is that the water remain for about 5-7 days, the time it takes for the larvae to go from egg to mosquito, so in places where the water from a rainfall dries up faster than that you aren't going to get a lot of mosquitos. If there's no still water, some mosquitos will lay their eggs in (slowly) running water as well, but few of the larvae are likely to survive as running water is likely to have plenty of other life in it and pretty much everything else that moves in water loves to eat mosquito larvae.

In order to keep the window of time that their larvae need to develop a small as possible, the females try to give their offspring a head start in the form of highly concentrated protein... the source for which is your blood! The only time they bite is when they're "pregnant" with eggs (so only the females), otherwise adult mosquitos feed mostly on nectar.