Much has been written about Absinthe but here is a quick primer to get you started.
Invented in 1792 by Pierre Ordinaire, a French Physician, living in Switzerland who was looking for a herbal mental tonic.
The distinguishing ingredient between most Pastis and Absinthe is the addition of Artemisia absinthium , a plant better known as grand wormwood or in America as European Wormwood, to a Pastis which is distillation/maceration mix of  herbs such as fennel, anise, melissa and hyssop among others. Recipes vary by country, manufacturer, and type. One of the many classifications of absinthe is by amount of alcohol or proof: absinthe suisse ,the best grade; alcohol content of 68-72%, demi-fine ,50-68% alcohol or ordinaire , 45-50% . Recently legalized (2007) after a ban in most European countries since the early 20th century it is produced mainly in France, Switzerland, Spain and the Czech Republic with widely variable price and quality.
 The two basic types are green (verte) and clear (blanche) Absinthe. The better green Absinthe is gets its color naturally from fresh absinthe (dried absinthe makes for a yellow color) rather than artificial means some of which are rather toxic although to a much lesser degree than around the turn of the century – the toxic additives and their symptoms may have helped get absinthe banned back then.  The clear is usually a double distilled Absinthe and is usually Swiss or  French in origin.
There is also a much rarer Rouge (red) absinthe that is made by the addition of hibiscus flowers to the herb mix. What you should look for is a balanced, complex, Absinthe that tastes good that you can enjoy. Otherwise what is the point ?

As to serving Absinthe, it is used in a number of cocktails the more popular ones being Sazerac and the Death in The Afternoon (Champagne and Absinthe Cocktail based on the Hemingway novel).

The most popular way of serving it and the most familiar to most is the Absinthe Frappe or Drip which can be an involved procedure if done with all the trimmings or as simple as a Scotch and Soda.

On no account should you ever  set fire to it ! This is a stupid and fake ritual created to sell the vastly inferior Hills Absinthe as a novelty sell. It is pointless, dangerous and a waste of good absinthe. It also leads to a lot of broken glassware and your dishwashers will hate you are the burnt sugar is almost impossible to remove from spoons and glassware.