Sex determining systems in animals work by flipping particular genetic switches (activating genes usually) which result in a cascade of events leading to the differentiation of an individual into one gene or another. The molecular genetics of many of these have been worked out in great detail.
The initial flipping of the "master" switch can result from:
particular environmental conditions (like temperature in alligators),
experimental manipulations,
or the counts of numbers of particular genes (genetic control).
Genetic controls like this can make use of particular genes on non-sex chromosomes, or on sex chromosomes (chromosomes that appear different from each other). Chromosomes are very large pieces of DNA which (often) carry many genes on them.
Sex chromosome systems have a homogametic and a heterogametic sex. The homogametic sex has two copies of the same kind of chromosome (like the XX in mammalian females), The heterogametic sex is the one with two different sex chromosomes (like XY in mammalian males). It can be the other way around, like in birds but the chromosomes are named Z and W and the female is ZW while the male is ZZ (maybe the Brits would say "double Zed").
Surprisingly, which chromosome is determinative can change in evolution. Recently, it has been demonstrated that genetic lines of
zebrafish (AB and Tübingen) picked up and fixed (only genes in the breeding population) significant sex determining genes mutations (probably when they were first generated from a small number of animals). Nevertheless, they still were able to make males and females, but not in predictable ratios.
Platies and swordfish (of the genus Xiphophorus) have changed their sex chromosomes during their complicated evolution (lots of crossing between different species).
There can also be weird and unusual (but natural) chromosome situations like YY or XXX. But they are not common.
