"I doubt ___" means "I don't think ___":

I doubt I'll be able to make it.

I doubt the weather will be any better tomorrow.

But you can't substitute "I doubt" for "I don't think" in sentences like this one:

I don't think he should do that.

In this sentence, "I think" is used to express your opinion. "I doubt" can only replace "I don't think ___" when it's being used to make a guess.

The phrase "I doubt ___" carries a greater level of disbelief than "I don't think ___". In other words, use "I doubt ___" for things that you really don't think are true.