Malaria has been infecting people long before there were people to be infected, and so has the genus Anopheles, A.K.A. mosquito, the only insect that is capable of harboring the malarial parasites that are capable of infecting humans. Malaria has been around for so long that it is possible that dinosaurs suffered from malarial infections. This long history has given the mosquito and the malarial paracites ample time to perfect their means of transmission, attack, and it's entire life cycle. It is hard to begin telling about a life cycle that it so perfectly continuous as that of malaria, so it is best to start with one of the transmission points. I will start here with the transmission to humans.
It is the female mosquito that transmits malaria, the male doesn't crave blood, but the female needs the hemoglobin in the blood, which it protein rich, to nourish her eggs. The bite isn't actually a bite, it is more like a sting, as her "mouth" is like a syringe needle. To keep the blood from coagulating, she uses a small amount of her saliva, which is when the paracite is transmitted. The plasmodium (scientific name for all malarial paracites) are single celled organisms that are so tiny, that 50,000 of them could fit into the period at the end of a sentence in a typical book or magazine. With the typical bite, less than one hundred of these organisms enter the bloodstream, but one is all it takes to kill a person. The parasites are only in the bloodstream long enough to get to the liver, which is where they begin their destruction.
The first stage, known scientifically as Sporozoites, is over when the plasmodium enters the liver cells. At that time, they transform into the Merozoite stage, where they multiply and eat the liver cells out. After about one week, when there is no more food in the cell, the paracites (around 40,000 for each one that originally entered the body) burst out of the liver cells and re-enter the bloodstream. There, the enter the blood cells within 30 seconds of re-entering the bloodstream. The malarial paracites then eat on their new cellular hosts and multiply for two more days, until they have exausted their new food supply again. Then they burst out and invade new blood cells in a continous cycle until the disease is treated. This cycle-within-a-cycle is when the person infected shows signs of malaria, the fevers and chills. Some forms of malaria, all within the Plasmodium Falciparum group, the worst forms of malaria to get, can cause brain swelling.
During this cycle, some of the Merozoites transform into male and female types called gametocites, which is how the mosquito picks them up when she feeds.
In the mosquito half of the cycle, the gametocites are now known as the gamites. In the mosquito's gut, they merge back together to form an ookinete (the first stage of an eggsack of sorts), the ookinete developes into an occyst (the eggsack proper), and eventually ruptures, and all of the Sporozoites (the first stage in humans as well) come out, to migrate from the mosquito's gut to it's salivary glands, which is where they stay until they are injected into another human host.
And the cycle begins again. This complex cycle takes only a month at most to complete, making trating and combating malaria more difficult, as the bugs can defeat a new tratment faster than the treatment can wipe out the strain.
Sources: National Geographic Magazine and Wikipedia