In some of my applications, they have this option. If I choose summer, am I REQUIRED to take courses during the summer? If not, then why would I ever choose fall 2015?
Yes, if you choose summer, they are expecting you to take summer courses. You cannot be accepted for summer then defer enrollment to the fall, they would just take away your acceptance altogether.
Many people see summer enrollment as being less difficult to be admitted under rather than fall, but that really depends on the school. So, if you want to start in the fall, apply to the fall.
If one intends to start in the fall semester, one applies for the fall semester. It’s much easier to be accepted for the fall and later decide to start in the summer (this is often arranged), than it is to be accepted for the summer and later decide to start in the fall (this is not so often arranged). I doubt that your chances will be different in applying for one versus the other. You presumably still intend to stay at the school for a length of time equivalent to obtaining a bachelor’s degree, and not just the summer. Thus, your application is still evaluated in the same way as any other student applying to the school.