Before getting into some applications with the capo, lets get some basics out of the way.
1) Put the capo as close to the forward most fret (bar) as possible without
disturbing your fingering. Put it on and play a bit to make sure it feels
good where you put it, and that you hear no "buzz" from the open strings.
2) Put the capo on "perpendicular". Make sure it is not angled. Also, make
sure it is on fairly tight. Some capos are adjustable, if so, make sure it is
on tight enough to secure good tuning. A moving capo may put the strings
out of tune.
3) Tune the guitar after you capo! The positioning of the capo will affect the
tuning. Of course this will depend on the guitar you are using and other
things as well. But, I would recommend you make a habit of this.
Kinds of capos:
There are several kinds of capos. I myself have a few. I primarily use the Shubb adjustable made of stainless steel (I use this mostly for recording sessions). I also use the Kyser quick capo (usually for "live" performance).
I also may use 2 capos (or have handy) for live playing if there is a particular key change. There are also "partial" capos, "cut" capos, and probably a whole bunch others!
Now, some applications:
The applications I will present go basically from simple, to more advanced.
So, if a particular application does not apply to you, move on.
If you are a guitarist who plays mostly solo (by yourself), or a beginner, the following may work for you.
For beginners, I might recommend you try a capo if your guitar is physically difficult to play. Often, a beginning guitar (usually a less costly guitar) has strings that are further off the fingerboard, especially in the open, and first positions (the nut of the guitar might be placed too high). Even some good guitars are made this way for better sound. When I taught private guitar (and my wife taught as well) we would recommend this to students (especially young ones) to try this before purchasing another guitar.