The pen tool allows you to put down a series of points that attach themselves with a line. You can tweak the tension of these points to change the curve of said line from point to point, so you can trace over linework. Since it's vector based, you can upscale pen lines without information loss.
To actually use it, go to the pen tool, and trace your object out with as few points as you think you'll need (I mostly place one around the tops of curves - it's hard to explain, but once you try it it'll be more obvious what you need to do). If you add too many points it takes forever to trace and ends up looking like crap.
After you finish making the line or curve you want, go to the pen tool on the sidebar and drop down the variation bar with that little tab, and select the bottom 'convert point tool'
Then click on the points individually and stretch out the handles. These handles influence both sides of the curve on either side of the point. Stretch them until they trace over the line you want.
When you're done, right click and select stroke path, and it'll emulate your brush along that curve.
You can also save pen tool paths you made, which is obviously very handy.
There's more stuff that goes way in depth, but those are the basics. It's actually really easy.
I don't understand these terms. What is inking?
Think of inking in comic books. You have the pencil linework, then you ink the lines in with, well, ink. Color is an option too.
This can more or less be done in PS and stuff with vectors and the pen tool and stuff. It gives really clean lines.