A graduate degree is a degree in which you must have previous experience or a previous degree in order to take. Most Masters or PhD courses are graduate degrees.

An undergraduate degree is usually the first degree people will get in college. It is a degree that requires no prior college experience. Basically an undergraduate is the degree type most people do after high school. Undergraduate degree include Bachelors in Arts BA, Bachelor of Science BSc etc.

While many people do an undergraduate degree people will usually only do graduate degrees if they are interested in research or their job requires it. Many teachers are required to have a masters in education for example.

(I am speaking in general there will probably be some exception to what I say especially depending on where you are from)