What is it?
In brief, the mezuzah is the little oojit on Jews' doorways; it's a wee scroll inside a protective case, also (confusingly) referred to as a mezuzah. The scroll is handwritten on parchment, in accordance with an awful lot of laws, in very tiny writing. Stam.net has a lengthy introduction to mezuzot; Wikipedia has a shorter but more wide-ranging article.
Where do you put it?
Here, stam.net explain where and how to hang your mezuzot.
Here, I give some examples of how not to hang your mezuzot.
Mezuzah Cases for the Artistically Challenged
It's nice to make your own. Here is my step-by-step guide: how to make a simple, fun, cheap, functional mezuzah case.
Mezuzot take time to do properly. This makes them pricey. If you buy a very cheap mezuzah, chances are it's not kosher because someone has taken shortcuts.
A LOT of invalid mezuzot are sold with a kosher stamp on them. Anyone can get printed a plastic bag with a kosher symbol. The mezuzot in these are scribbled, often with letters made invalidly. Sometimes they are written by little boys or non-Jews because that is cheaper. Sometimes they are written on parchment which has been treated to make it easier to write on - but which will decay in a few years.
In short, if you're going to buy mezuzot, buy good ones; it'll be a saving in the end. A $15 mezuzah is basically $15 wasted, and you probably won't even be fulfilling the mitzvah.
I suggest stam.net for general mezuzah buying.
May a Woman Write A Mezuzah?
Here is a brief analysis of the issues specific to mezuzah.
Buying Mezuzot from Me
Not everyone agrees that mezuzot written by women are kosher. If you are buying a mezuzah from a soferet, make very very sure that the recipient is okay with that. Never assume that someone is aware of the issues. I have had very learned rabbis assure me that there is nothing wrong with a woman writing a mezuzah, and they were very surprised when I showed them what the classical law compilations say.
More information about buying from me here.
Jen Taylor Friedman, 2008