AKA the Fisher-Muller Model
Muller's ratchet is a hypothesis developed to help explain why sexual reproduction might be more beneficial than asexual reproduction.
In asexual reproduction, genomes are inherited unchanged from the parent generation. This means that once a mutation enters a family tree1 it will stay there, to be inherited by every descendant. This is not true in sexual reproduction, as any given gene may be absent in any given descendant (as long as one parent does not carry the offending gene). Even if a gene pool becomes very polluted by a counter-productive gene, a few unaffected individuals can spread an 'unpolluted' gene through the entire population, given enough time.
In an asexual population random mutations will slowly collect, and the great majority of these mutations will be harmful. This collection of mutations is known as genetic load, and to some extent this burden exists in every population of every type. Asexual populations have no obvious corrective defenses to help combat the collection of harmful mutations, and therefor might be expected to be slowly but continuously working their way towards extinction.
Biologist Hermann Joseph Muller likened this constant, unidirectional2 descent into decrepitude to the movement of a ratchet3, a device that will only allow movement in one direction. While it was originally intended to describe, in carefully modeled terms, one way in which sexual reproduction is superior to asexual, it has developed a slightly different focus. It has been found, over and over again, that populations that were once thought to pass genes on without any means of recombining DNA have developed interesting tricks to mimic sexual reproduction. Bacteria, viruses, fungi and other single-celled organisms can trade genetic material through transformation, conjugation, transduction, and the parasexual cycle. This means that very few beings on this Earth are subject to Muller's ratchet as originally conceived. However, Muller's ratchet does help explain why these organisms have evolved alternative methods of trading genetic material.
Muller's ratchet is an area of active research, as we do not see its effects to the extent predicted in those organisms that do not undergo recombination of any sort. Some species of lizards and insects, for example, reproduce entirely through parthenogenesis; likewise, mitochondria and chloroplasts do not engage in any known form of genetic recombination. It is expected that Muller's ratchet does play a role in these populations, but at this point comparatively little is known about mutation management in these organisms.
Other reasons to have sex include the Hill-Robertson effect, minimization of genetic hitchhiking, and the tangled bank hypothesis. These are contrasted by the two-fold cost of sex and other costs of sex.
1. Or, as might be more apt with asexual reproduction a family 'pyramid', as there will be no branches, just a slow widening of the gene base as mutations collect.
2. There is always the remote possibility that a mutation will undo another mutation, a phenomenon known as back mutation. This is rare.
3. Muller was the first to use the analogy to a ratchet, but the term 'Muller's Ratchet' was coined by Joe Felsenstein in his 1974 paper, The Evolutionary Advantage of Recombination.