More specifically, what methods exist to cool your wort after the boil, and what are there respective pros/cons?
There are a few methods, each with advantages and drawbacks.
Cool your Kettle
Take your hot brew kettle, full of wort, and submerge it in ice water.
Advantages: No real equipment expense, just take your pot and put it in a cooler or bathtub full of ice water.
Drawbacks: Extremely slow, higher risk of contamination. Probably have to change out the ice several times.
Mix in Ice
Add ice directly to the hot wort. (Alton Brown does this in his brewing episode.)
Advantages: Same as above.
Drawbacks: Can only be done for a concentrated wort boil. Only able to add enough ice to get to desired volume. No way of really ensuring your ice is sanitary--so a significant risk of contamination
Immersion Wort Chiller
A coil of copper hose immersed in the hot wort, with cold water flowing through it.
Advantages: Much faster than previous two methods. Faster cooling means sooner yeast pitching, which lowers your risk of contamination.
Drawbacks: Modest equipment expense, plumbing setup required to route water into and out of the copper tubing. Chiller must be sanitized before use. Corroded copper can impart off-flavors to your beer. Still have to transfer cooled wort to your fermenter.
Counterflow Wort Chiller
A coil of copper hose within a coil of other (typically rubber, vinyl, or garden hose) tubing. Wort flows through the copper tubing, and cold water flows through the surrounding hose in the opposite direction.
Advantages: By far, the fastest cooling method. Hot wort goes in one end of the chiller, and by the time it comes out, it is cool. Chiller can be used as part of the siphon path on the way from brewkettle to fermenter. It's possible to cool a 5-gallon batch to pitching tempurature in less than 5 minutes.
Drawbacks: Also the most complicated setup. Interior of copper tubing must be sanitized. More expensive (usually) than immersion chiller. Requires reasonably extensive plumbing setup (siphon to chiller, chiller to fermenter, faucet to chiler, chiller to drain), and requires siphoning hot liquids. Requires cleaning the interior of a (usually) narrow copper tube when you are done.
Plate chillers and other heat exchangers
(Thanks to Morgan for pointing this one out)
Similar in function to counterflow chillers. Hot wort and cold water in, cool wort and warm water out.
Advantages: Same as counterflow chillers, possibly more efficient, depending on your model.
Drawbacks: Again, largely the same as counterflow chillers: expense, plumbing setup, and cleaning.
My first two batches used the "Cool your Kettle" method. It works, but it can take half an hour or longer. I've been using a counterflow chiller ever since, and I'll probably never use anything else.
Passive Cooling: I seal up the wort and leave it overnight in a room tempurature area or outside, whatever. If the area is too cold, I check the wort starting at about 6 hours, so it doesnt get too cold to pitch.
The only con I am aware of is the time it takes, but I bet it affects the flavor somehow, too. Anyway, I've won a few awards just doing this, so its ok with the styles I have made (ales).
The pros are obvious. Less work and less mess.
I brew 23-litre batches split between two 20-litre canning pots. Cooling them down in a bathtub of cold water works quite well.
I had an old copper immersion chiller, which a friend had gotten from someone else. It was old and a little tarnished. I cleaned it off to what seemed to me to be an acceptable level and used it on my first all-grain batch, and I'm pretty sure it ruined it. That batch is not drinkable, and the off-taste it has seems metallic.
But I think bad experiences from copper-coiled immersion chillers are very rare -- lots of people use them without problems. But now I'm happy with my bathtub solution, and don't intend to change any time soon.
Before or during the boil I usually set a few pots of water in my freezer and use that chilled water (and a bunch of ice packs) to fill my laundry tub and perform the "Cool your kettle" method. It's not the fastest method but using the chilled water helps a bit and I haven't had any bad experiences yet.