I'll is a contraction of I will. These types of "apostrophe words" are called contractions (though be aware that there are other reasons to use apostrophes besides forming contractions). The apostrophe indicates that one or more letters were removed, thereby shortening, or contracting, the word. In this example, I (wi)ll = I'll

As kb90 mentions, contractions are generally considered less formal, at least in writing. However, they are extremely common in speech—native English speakers use them all the time without thinking about it. In fact, they are so common that it often sounds strange, forced, or excessively formal to say "I will" in a casual conversation.

I agree with kb90'suggestions for use in writing; I would only add that "I'll" is frequently used in speech. While you will certainly be understood if you say "I will", your speech might sound a little more natural if you pay attention to and imitate the way that native speakers use contractions like "I'll".