What's for breakfast? is the correct way of saying it. If you ask "What is breakfast?" then maybe someone will give you the definition of the word "breakfast."

What's for breakfast? = What are we having/eating for breakfast? "What is breakfast?" sounds like either a philosophical question, or that you don't know the meaning of the word "breakfast".

What is breakfast ? The actual meaning of Breakfast is the first meal taken after rising from a night's sleep, most often eaten in the early morning before undertaking the day's work. However, I have seen some native English speakers claim that "breakfast" can be used to refer to this meal or to refer to a meal composed of traditional breakfast foods (such as eggs, oatmeal and sausage) served at any time of day. In US, some restaurants offer 24 hours breakfast. I believe, it is the main cause in changing its meaning from early morning into anytime of day.

Peachy is correct, by omitting "for", you are asking for the meaning "of" breakfast." Anyway "breakfast" means to break your fast (thus the first thing you eat in a day), thus it's not limited to occurring in the morning, as everyone eats at different times (hence all day breakfasts available at restaurants). The reason you say for, is to indicate what you would like to eat in order to have breakfast. So you are eating 'for' the purposes of breakfast. What are you using for a computer these days? (what are you using as a computer these days?) What do you want to eat "as" breakfast"? It translated out means "what do you want for the purposes of breakfast?"