What's the difference between "first", "last", "close", and "strong"?

I set things that are supposed to hit blimps to Strong. For example, a 4/2 Sniper has the Cripple MOAB ability. Or the */3+ Bomb Tower has a bonus against blimps. A Glue Gunner with Corrosive Glue set to Strong will be more effective. If position matters, things set to Strong tend to go towards the beginning of the track.

I sometimes set things that are supposed to fire down a long straightaway to Last. The 4/* Dart Monkey (Juggernaut) is good like this, as is a 3/* Super Monkey. The key there is their high pierce. This can also be used against lead Bloons. They are so slow that they are often last or close to it.

I've rarely used Close. I suspect that it exists mainly for things like the 3/* and 4/* Monkey Ace, which use it for targeting. Occasionally I've microed it to hit a specific target for some reason. You can read more information about using Close at Is targeting “closest” or “last” ever useful?

First is of course the default. It's especially useful towards the end of the track to make sure that the Bloons that are farthest along get hit. A 4/2 Ninja Monkey is effective set to First.

Boomerang Monkeys are an example of a tower that can benefit from different settings based on position. Basically, you want the boomerang to hit either the first or the last Bloon in a line and then get the rest. Since the boomerang always circles counterclockwise, depending on the track you might find First or Last to work better.

The 4/* Monkey Village can work well set to either Strong or First. Used with Strong, it will help crack blimps so that other towers can take care of the regular Bloons.

I usually have towers that are set to First and others set to Strong. I play a lot of games without any towers set to Last or Close, although they do have their uses. If you are microtargeting with a tower, you might select any of them because it happens to target the Bloon you want to hit. And of course, there are always one-off situations where an odd position can encourage an odd targeting choice.

Just to add to Brythan's excellent answer: One tower that benefits from using Last is the Boomerang Monkey. It takes a lot of practice to envisage the arc that a Boomerang will traverse, but there are some places where you can get excellent amounts of damage as long as the monkey is firing along the track in the right direction. Usually this is easiest to ensure if the monkey fires towards the direction the bloons come from, which happens most if you use Last mode.