Copic is a brand name for a range of Alcohol ink markers. The most popular types of alcohol marker used by crafters are Copic or Letraset ProMarkers, but there are other makes on the market. There are other types of marker with water based inks or other mediums.
The Copic alcohol markers come in several different shaped barrels, but the ink is the same inside each. They are given names and numbers so you can match up across the range. The most popular are Sketch and Ciao pens. Ciao are the cheapest so good to start with, they are the easiest to find here in the UK but I think in the USA the Sketch seem to be more readily available.
If you want to try before you buy kind of thing, then take an introduction to Copics workshop - you usually get a few pens along with other goodies and will be shown how to blend the colours etc. If you are already experienced with colouring and understand the colour wheel and how to blend you might not find it so helpful, in that case just get a couple of pens to try, or maybe borrow from a crafty friend. It does take practice to get good results, but there are a lot of tutorials on line with helpful tips and advice.