Only female mosquitoes bite. They usually require the protein from their victim's blood to produce eggs. In some cases, females may develop eggs from nutrients, etc. With one blood meal, a female may produce as many as 250 or more eggs. After a blood meal, it takes 3-4 days for blood to be digested and the eggs to develop.
Females transmit diseases when they live long enough to spread viruses, microfilaria, etc.
Only a very small percentage of mosquitoes live this long. Larvae or wigglers develop through four instars in 10-12 days and change into pupae or tumblers. Pupae do not eat but metamorphose into adults in a few as 1-2 days in the summer. Females seek a place to lay their eggs, and the site chosen is probably based on clues from chemical receptors on their body. Mosquitoes will either lay eggs on existing water bodies, or they will lay eggs at sites that will later be flooded.