I understand that one of the main reasons to use different kinds of fat or oil is the temperature at which it burns, e.g. an extra-virgin olive oil to fry a steak wouldn't work. Taste is also a very important aspect. Can you give a list of the most commonly used oils/fats and what they are used for?
"Commonly used" depends mostly on the culture, I'd assume. There's a lot of different oils, so I've organized by use rather than try for a complete list.
Some of the ones that you might find in a "typical American" foodie's kitchen include:
For frying: something with a high smoke point : peanut, sunflower, soy, extra light olive oil
For baking (muffins & cakes): something with a mild flavor : corn, canola, "vegetable", soy
For baking (biscuits & pastry, or greasing a pan) something solid at room temp : butter, shortening, lard
For general pan cooking: olive oil (any kind), butter, anything from the "baking (muffins)" list.
For salad dressing: any nut oil, mild oil, or virgin / extra virgin olive oil
For sauces: Butter.
For finishing: something flavorful to drizzle over at the last second... odds are, it's extra virgin olive oil, but possibly sesame or a nut oil.
Now, there's regional differences -- in the south, it's pretty common to save your bacon grease for cooking and to use shortening for frying. Lard's still popular in hispanic (and likely other) cuisine, schmaltz (rendered poultry fat) is used in both Jewish and French cooking. Ghee (similar to clarified butter), is used Indian cuisine ... and the list goes on.
If you're looking for a 'must keep on hand' list -- a mild oil, extra virgin olive oil and butter will get you through most anything. Add shortening if you like baking, and sesame oil if you like to cook asian food, and you'll be prepared for most anything.
There are a great many oils and fats on the market, which you choose to use will largely depend on several factors:
The most common oils are probably
Olive oil has numerous health benefits and is great for the heart. Olive oil has a smoke point range between 208c for Extra virgin and 243c for Extra Light.
Sunflower oil - A good all purpose oil, useful for cooking and salads. Sunflower oil has a smoke point of 226c
Corn oil - Not a great deal of taste but is great for frying. it's also not terribly healthy. Corn oil has a smoke point of 233c
Peanut oil - My favourite when cooking Asian food. Peanut oil has a smoke point of 225c.
Butter is also commonly used in cooking either alone or with oil. Butter brings a richness to sauces, it's also great for adding at the end of the cooking process to add a little sweetness and shine to the food. Butter has a smoke point of 150c.
Another form of butter, referred to as clarified butter or ghee is used in the preparation of Indian food. it has a slightly nutty flavour and a high smoke point. I nearly always make my own but you can buy it. Ghee has a smoke point range between 190c and 250c.
It addition to the aforementioned oils, there are a number of 'specialist' oils that are fantastic in salads:
All of these have a very distinctive flavour and should be used in moderation. sesame oil is really great in Asian food. Add a little sesame oil with the peanut oil for cooking or add a little towards the end.
Edit: I forgot to mention lard. Lard is used in a variety of cooking processes, including baking and frying. It has a relatively high smoke pint, which makes it ideal for deep frying. It's also used in a process called larding, where small amounts of lard or lardons from bacon fat are injected into lean meat by means of a larding needle.