"I have to" means "It is necessary". It is similar in meaning to "I must".

"I'm supposed to" means "I'm scheduled to" or "I'm expected to" - in the sense that this is what other people are expecting you to do.     

These two ideas can often be in conflict with each other. Here's an example:

"I have to lie down and rest. I'm supposed to be working this evening, but I can't. I'm just too tired."

I have to lie down = This is what is necessary

I'm supposed to be working = This is what I'm scheduled to do  (but I'm not doing it)

be suposed to equal think.