Hemoglobin is the oxygen-carrying protein found in the red cells of our blood. Patients with sickle cell anemia have a sickle form of hemoglobin. While healthy red cells are shaped like a disk pinched in the middle (a doughnut without a whole), in patients affected by the disease the sickle hemoglobin forms strands which cause the red cells to be shaped like a crescent. This causes all sorts of complications and risks: the sickle red cells are more rigid and increase the risk of ischemia and necrosis; the drop in hemoglobin can cause painful enlargement of the spleen, anemia, tachycardia.
Sickle cell anemia is caused by a single-base polymorphism, SNP rs334, in the hemoglobin gene. The "normal" allele is A, whereas the mutated one is T. As you know, we all carry two copies of each genes, so the vast majority of the population has AA at this locus, a few carry AT and very few TT. Heterozygous carriers of the SNP (people who carry the AT form) are, for the most part, not affected by the disease. The healthy gene allele produces enough healthy hemoglobin to compensate for the sickly one.
This is not uncommon in genetics: another example of disease that's only expressed when both gene copies are mutated is cystic fibrosis. You may carry the "mutated SNP", but so long as you have only one copy, you're not affected by the disease. The problem is when both parents are carriers: if the child inherits both mutated alleles, she will develop the disease. Now the question is: if this allele causes such a devastating disease, why has it not been wiped out by natural selection and/or genetic drift?
The answer lies in a concept called heterozygote advantage: if the one allele turns out to be advantageous in certain circumstances, then it will still be prevalent at a certain frequency in the population. For example, the allele that causes cystic fibrosis has been hypothesized to protect against cholera, typhoid, and diarrhea. In the case of RS334, the sickle cell anemia SNP, the mutant has been proven to confer an advantage against malaria:
"These deleterious mutations are maintained in the population in a state of balanced polymorphism because of the protective effect against severe forms of malaria conferred by the heterozygous states. [...] However, it is not clear what happens to these polymorphisms in areas where the selective pressure has become relaxed or is nonexistent altogether. In some cases, as in the Caribbean, the HbS mutation has been shown to continue to exist with unaltered frequency, despite the near eradication of malaria more than half a century ago .In  Salih et al. looked at the prevalence of malaria in the population from two African villages, Hausa and Massalit. The two populations were of different ethnic origins, and in both malaria was endemic but mostly mild. Within the two populations, they looked at the genotype frequencies of the sickle cell polymorphism and used a simulation to predict its behavior. Although the sickle cell SNP conferred significant protection from malaria in both populations, they found a trend for a decrease of the RS334 allele frequency in Hausa and an increase of frequency in Massalit. They conclude that this effect may be due to the fact that
"In the Hausa village, this seems to be likely due to the low clinical burden of the disease, the population effect (possibly under drift) in addition to the deleterious impact of the homozygous allele, both conspiring against the maintenance of balancing selection. In the Massalit village, the relatively higher episodes of clinical malaria, in addition to a potential impact from visceral leishmaniasis (a disease with a higher fatality rate), may be responsible for the different selection profile."
Salih NA, Hussain AA, Almugtaba IA, Elzein AM, Elhassan IM, Khalil EA, Ishag HB, Mohammed HS, Kwiatkowski D, & Ibrahim ME (2010). Loss of balancing selection in the betaS globin locus. BMC medical genetics, 11 PMID: 20128890