Does that expression mean "I'm totally sure" or "I'm almost sure"? I always thought the first option was correct, but once I saw in a series episode a character who said that, and his interlocutor expressed skepticism about it (something like "Are you sure or pretty sure?"). Does that depend on context, or does it always mean the same?
We can't assign a percentage of certainty to it, but it isn't close to 100%. Perhaps 60-85% certain is close to what it means for most native speakers. The more emphatic the statement (stress and pitch and volume are involved, because this is primarily spoken English, but in written English it might be underlined or in bold font or in capital letters for emphasis), the more certain the speaker/writer is. I'm quite sure probably means about 90-100% certain, however.
I'm going to go ahead and propose a different answer, despite @Bill_Franke's answer already having more than 10 upvotes. I think Bill's answer is likely a perfectly correct representation of the usage of pretty sure in American English but I speak British English and to me it means something different.
a. Qualifying an adjective or adverb: to a considerable extent; fairly, moderately; rather, quite. In later use also: very.
Whilst I would understand the to a considerable extent meaning, given context and intonation, my default would be to assume this later use of very. Further, if I were to say she's pretty good looking or, for our current question, I'm pretty sure I could drive that van, these would mean she's gorgeous or I can definitely drive that van, no question.
b. pretty much: almost, very nearly; more or less; (also, in early use) very much, considerably.
Whilst this definition is specific to pretty much, I see it as very synonymous with pretty sure. The phrase I've pretty much finished work would imply, to me, that only a few moments' work is left. The phrase I'm pretty sure Adam finished work by now means that I would be very surprised if Adam had not finished work.
Finally, pretty sure can be used as a form of politeness or disingenuous courtesy. I'm pretty sure that's not how you do it may be seen as a nicer alternative to Wrong!, and I'm pretty sure you'll find that's not the case is somewhat less friendly, though more socially acceptable.
In brief, I believe that in British English, pretty sure would generally be taken to mean certain, or at least 90% close to certain. I currently live in the USA and this is often a source of confusion.
So much depends on context. We may also need to take irony and a feel-free-to-see-for-yourself attitude into account.
I wouldn't sit on that bench if I were you. The paint is wet.
-- Looks dry to me.
I just painted it, so I'm pretty sure it's wet. But go ahead, have a seat.
The phrase "I'm pretty sure" there, while it means on its surface "There's a chance I'm wrong", actually means "I know damn well that the paint is wet but it's ultimately up to you to make that determination for yourself if you're the kind of person who ignores a friendly warning when you're given one."