There are two types, rough and smooth. There is also everything in between so there is actually a lot of variety. My first bc was a very smooth looking rough and when she had blown coat, could easily pass for a smooth.
All of mine have been cattle working lines and those are mostly short coated.
There is the occasional rough coat in several lengths and with more or less undercoat.
For conformation, the ones I have seen in our performance dog club, that were winning plenty also in conformation shows, were all very heavy rough coated, with a beautiful full coat when showing.
I don't remember seeing a real smooth coat win in conformation.
In NZ - "Rough Coat Collies" (Lassie type) are not Border Collies - related but not the same. That had me a little confused at first.
In NZ - we talk of smooth, medium or long coat. We then mix this up by describing how the coat lies regardless of length - essentially smooth or "fluffy" as well. "Fluffy" is not deemed to be correct but a lot of our farm-line BCs have it. My older BC is a medium coat and fluffy. (Nicest nickname is Miss FluffyButt - but she goes by the "paddock" name of "Grub" so it is an improvement .). My younger BC is a smooth medium coat - meaning that her medium length coat is smooth and lies flat. "FluffyButt"'s coat certainly doesnt lie flat except when she is saturated - even then it very quickly dries into curls with that added bounce.
AKC states there are two varieties: smooth and rough.
Looks like we can't reach a clear decision. Glad I'm not the only one scratching my head. A friend recently got a (agility) pup and invited acquaintance to bring her flyball pup over for a play date. Flyball lady looked at agility lady like she had three heads and horns when she asked which, of two, coat types flyball BC had.
I understand there is a lot of variety. I guess the contradicting coat descriptions is a casualty of war between AKC and ABCA.