Balancing selection
The sickle cell allele is a mutant form of the gene coding for haemoglobin, the oxygen-carrying pigment in red blood cells. Given how serious sickle cell disease is, why does this allele persist?
The answer lies in selective pressures favouring the 'abnormal' allele: although having two copies of the allele (one from each parent) is bad news, having one can be an advantage – it protects against infection with the malaria parasite. So in regions where malaria is present, there are 'balancing' selective pressures at work. The losers are the children unlucky enough to inherit two copies of the sickle cell allele.
Old age
Harmful alleles can also survive if they act late in life, as in the case of Huntington's disease. The defective allele doesn't show its effects until later life, after the age of reproduction. As it does not affect someone's reproductive success, natural selection cannot act upon it.
It is possible that other alleles have negative effects in old age while being advantageous when we are young.
Good gene, bad gene
This highlights the importance of context. A 'bad' gene or allele of a gene may only be bad for us because our circumstances change. Some alleles seem to predispose to obesity or cardiovascular disease by altering our metabolism. This may be bad for us now, but in the past it might have been an advantage – with resources scarce, being able to use fats more effectively might have been very handy.
Finally, although there's still debate about this, natural selection might not have 'caught up' yet. For instance, the gene causing cystic fibrosis may have protected against cholera – but now that cholera is no longer a problem in many countries the gene no longer seems to have any advantage.
Most harmful alleles, though, are quickly lost - a process known as purifying selection.