Let's start with us humans - as you know, humans with two copies of the X chromosome are female, and those with two different ones (XY) are males. But what if you are missing an X or a Y chromosome? What if you have an extra sex chromosome? XYY individuals aren't "super men", but XXYY (or XXXY or XXY) individuals are sterile males. Women with an extra X chromosome (XXX - trisomy X) are developmentally delayed, so there is such a thing as being "too much woman". But missing an X chromosome results in Turner's syndrome for women (XO), and spontaneous abortion for males (OY).
So what does this tell us about these chromosomes and sex determination? Well, you HAVE to have the Y chromosome to be male, and an X chromosome to live, but having more X chromosomes doesn't make you more of a woman - it just makes you sicker. This is because the X chromosome actually carries a lot of important information - including sperm production - while the Y chromosome has been shrinking for the past millions of years and pretty much just carries a single important gene. This gene, SRY (pronounced "sorry"), is a regulator for testis development early on, but all the genes controlled by this are on other chromosomes. Thus came the (in)famous statement that y chromosome shrinkage may be driving men extinct - although this has been disproved. One reason for this is because the SRY gene alone can determine maleness, and it is possible for it to insert into the X chromosome (XX males).
Look how tiny the Y chromosome is compared to the X chromosome! Image from: fairbanksirl.com
But what about other organisms. That fly buzzing over your half-rotten bowl of fruit you haven't quite managed to finish? Femaleness is decided based on having two X chromosomes, rather than maleness being on the presence or absence of a Y chromosome. Those cockroaches crawling out of your walls? They only have X chromosomes, with males having a single copy and females having two. And that sparrow outside your window? It is like the opposite of humans - men have two chromosomes the same while females have two different ones.
But perhaps the most intriguing method of sex determination is found in sea turtles (and alligators), whose sex is determined by egg incubation temperature. Hot eggs are female, and cooler eggs become male; the temperature difference is very slight, and therefore a mother can somewhat control the sex ratio of her offspring by rearranging eggs. However, some researchers worry that this temperature differential will not be possible under a future climate, and so the reptilian world will be run by females. Of course, females are generally better at spacing sexual encounters than males, so a female-shifted population may not be all bad and could mean more sea turtles.
How many methods of sex determination can you count? From https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcSARbq3hpD6h4wLG7fTv7Z_wRnGZBIr1ttjUeX0qeoTjz1u9aW7Fw
Or what about organisms which can reproduce without sex (an administrator at creation.com denies Jesus was conceived this way). Or those which switch sex at some point in the lifecycle (think Nemo). Or earthworms, which just serve as whatever sex they feel like, and usually have sex with another worm, but can also just fertilize their own eggs.
There are so many more organisms - like sharks (thanks Rebecca!)- that we know nothing about. Understanding sex determination is more than just an intellectual question - it has important implications for managing wildlife, whether it be on the endangered species list, our menu, or a novel invasive species.