How do the functionalist and conflict explanations of stratification differ?

According to functionalism, categories like class, gender and race fulfill certain social functions. This can be cultural or it can be structural.

For example, a working class person has that role because it is functional for the society as a whole. There will always need to be workers just as there will always need to be employers. Society consists of more or less fixed roles that people slide into. Likewise, gender categories are there because they fulfill necessary social roles. Women have specific jobs to do and if they do not do them then society as a whole suffers. The upshot of this is that people should learn to appreciate the way society works as it is now and not try to ruin the equilibrium through radical reform.

Conflict theory (and the best example of this is probably Marxism) stresses the antagonistic interests between these categories. Rather than functioning as a beneficial social whole, classes, genders, races mean that some people are exploited for the benefit of others.

For example, Marxists don't think that class divisions are functionally necessary. They think that the only reason class exists is because it benefits employers (capitalists) who live off the surplus wealth generated by workers. It is not in the worker's interest to continue this arrangement, rather it is in their interest to overthrow the economic system and abolish classes altogether.

Feminism might also be called a conflict theory because it sometimes posits that the interests of men and women are mutually antagonistic. Any progress toward gender equality must occur at the expense of men, so the conflict is inherent. There are racial theories that reflect this too, such as the Black Panthers and Malcolm X, etc.