There are a lot of ways that sink drain plumbing parts can fit together. Different sinks and configurations will require different parts.
If you have the dreaded clogged sink see 'How To Unclog a Drain'.
It is impossible to cover every possible situation. Instead we will try to discuss the various parts, their purpose and how they fit together.
There are many variations of sink drain parts. We will try to address the common ones.
P Trap - This is a two piece fitting that makes the trap. Water stays in the bottom of the trap to keep sewer gases from getting into the house.
S Trap - Similar to the P Trap the S Trap is used when the drain is in the floor instead of the wall.
Tailpiece - This is the piece that is connected to the strainer on a kitchen sink. For a bathroom sink the tailpiece and pop up assembly are all one piece.
Tee - Tees are used for double sinks, usually in the kitchen.
Waste Arm - The waste arm connects the tailpiece to the tee on a kitchen sink. You will usually have two of them.
Elbow - Used sometimes on odd configurations. Most of the time this piece is not necessary.
Extension Tube - Used to extend the tailpiece or other pieces that are too short.
Coupler - Used to connect two pieces of drain pipe together. Usually not needed but useful if you cut something too short.
Sink Strainer - The hole in the bottom of a kitchen sink is filled with a sink strainer. This piece seals the opening and directs the water into the drain pipes.
Pop Up Assembly - This assembly is used for a bathroom sink.
Before you disturb anything you want to take a picture of your configuration or make a sketch. Take some measurements, since many of the items come in multiple sizes. When you go to purchase parts you will want to take your picture or sketch. Try to identify each of the parts you have so that you can get the right ones.
Plastic fittings are definitely easier to work with. However, you do not want to mix them. For homes with chrome or brass fittings you want to use metal replacement parts. Unless you are replacing everything and then you can switch to plastic. Already have plastic? Great, no problem at all.