Oils and fat are categorized into hydrogenated, unsaturated, monounsaturated, polyunsaturated and saturated forms depending upon their chemical structure. Which you select will have an effect on your cholesterol level, for example.
For flavor and health, many people prefer olive oils. There are varieties such as light (little flavor) to extra virgin (darker, richer flavor). The type you select often depends on the ultimate use (light for baking, extra virgin for salads for example).
Fats can have a different heat tolerance, which can impact their use in frying. Most oils such as vegetable, corn and canola are fine for home kitchen frying. Corn and peanut oil are often called for in certain recipes. They impart a light flavor and are used in some traditional recipes.
Solid vegetable shortening is prefered by some for frying and baking. It reaches a high fry temperature and leaves items crisp and "dry" (as my Mom used to describe).
Which fat or oil you use in baking will have a great impact in certain products, especially cookies and pie crusts. The texture may be from soft and moist (margarine) to crisp (shortening) even using the same recipe (I did this test for work once).
Here are a couple websites with basic info:
http://www.healthcastle.com/cooking-oils.shtml
http://www.healthycookingrecipes.com/cookinghealthyarticles/canola-cooking-oil.htm