"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.