Evolutionary elimination of recessive gene
Simply put, there may not be enough selective pressure to force those alleles out of the population. For example, take a hypothetical gene for eye color. The dominant allele gives brown eyes, the recessive blue, due to the lack of expression of a certain pigment-processing enzyme. So, two heterozygous (brown-eyed) organisms are crossed:

   |  A |  a |
--------------
 A | AA | Aa |
--------------
 a | Aa | aa |
--------------


which results in 75% brown-eyed offspring, and 25% blue-eyed offspring. Unless there is some very strong factor in the environment that has a negative selective pressure on blue eyes, the recessive allele will remain in the population.

OK, that's for an allele with no negative pressure acting on it. What about a disease allele, for example one that has a point or other mutation inactivating an essential enzyme, such as iduronate-2-sulfatase (I2S), a key enzyme involved in the breakdown of waste in the lysosome. It is part of a group of diseases known as lysosomal storage disorders, and is the causative agent of Hunter Syndrome. 

People who have the so-called "wild-type" allele(s) of I2S have normal function, even if they are heterozygous. However, if two heterozygotes produce offspring (or if a heterozygote produces offspring with a homozygous recessive individual), then there is a chance of a child with two recessive or defective forms of the enzyme, and they are much more likely to develop Hunter Syndrome (almost no genetic disease is 100% penetrant). If untreated, this disease can have many quite horrible effects (especially given that it usually appears in childhood). The disease is also progressive (it gets worse the older you get), and while it in itself does not typically cause death, patients have (sometimes severely) limited lifetimes.

So why haven't the mutant, disease-causing forms of I2S been "weeded out" by natural selection? In a single word: heterozygosity. Studies (I don't have a reference, unfortunately) have shown that supplying a Hunter patient with as little as 10% of the typical amount of active enzyme in an unaffected person is enough to resolve the acute symptoms of the disease, and begin reversing many of the long-term effects. All things being equal, one would expect that a heterozygote carrying one functioning allele of I2S and one mutant allele would express about 50% of the amount of enzyme that a completely unaffected individual does, and this is more than enough to prevent the onset of disease. There are likely other molecular mechanisms involved to express the functional version even more, so their levels may be even higher than 50%.

So, while there is certainly a strong negative selection pressure on people carrying two copies of a mutated allele (or two different mutations, there are actually many of them) that prevent most or all expression of active protein, that same selection pressure does not apply to heterozygotes, and so the mutant alleles remain present in the population.

I strongly recommend reading through the fairly short but definitely thorough course Understanding Evolution from the University of California, Berkeley. It may explain some of the concepts I've mentioned more clearly.

Note: I once worked for Shire, which, among other things, specializes in rare diseases including lysosomal storage disorders. I specifically worked with their Hunter Syndrome therapeutic, known as Elaprase. Other companies are also working on various therapeutics.

This is in no way an endorsement of the company or its products, it's just where I learned all of this stuff.