The most important thing about frying foods is temperature control. When oil is the proper temperature, it will flash evaporate moisture on the surface of the food, which creates a barrier that prevents the oil from penetrating the food and making it greasy. If the oil is too cold, this barrier won’t exist and the oil will seep into the food, making it greasy.
On the other hand, if the oil is too hot, the outside will cook faster than the inside, potentially burning the outside before the inside comes up to temperature.
Some recipes will call for specific oil temperatures to achieve specific outcomes (some classic frites/French fries recipes call for two-stage frying, first cooking at 325° and then a second cooking at 375°), but a general temperature for frying is 375°F.
Not all oils are good for frying. Even when frying at 375°F, you don’t want to use an oil with a low smoke point (the temperature at which the oil starts to smoke and become unappetizing). Unrefined oils, such as extra virgin olive oil, contain a lot of minerals and enzymes that break down at lower temperatures, thus giving the oil a low smoke point. More refined oils have fewer of these impurities, which gives them a higher smoke point (and generally makes them more neutral in flavor). Safflower oil is the best, but a good quality vegetable oil will work for most applications.
Used oil can be filtered through a fine mesh sieve and reused several times before it starts to break down, or take on undesirable odors/flavors.