Earphones often feature different sizes and types of ear cushions to achieve the most comfortable fit possible. Examples of cushions include memory foam, rubber and silicone. Don't let their small sizes fool you - earphones can get into the extremely high end of the price and performance spectrum. For example, the IE 80 in-ear headphones from Sennheiser cost around $450 (as of November 2012).
Ear buds usually do not have cushions (although they can), and they are often less expensive than earphones and in-ear headphones. If you want something for the gym that you don't care if you step on them on the treadmill, or if you need the thirty-umpth pair for your teenager, ear buds are your friends.
Here's a list of cheap ear buds that have good performances for their price points.