The short, but possibly frustrating answer is — whatever kind(s) you agreed upon. This should be outlined in a contract, up front, so you are both clear on the expected deliverables.
I’d recommend the following:
- Vector file (.ai or .eps file extension)
- JPEG — small, medium and large versions
- PNG — small medium and large versions
For each of the above, you should have positive and negative versions (for placement on light or dark backgrounds, as well as an all-black and all-white. You should have an RGB version (for screen) and a CMYK version (for print).
This may seem like overkill, but I’ve seen too many clients lose their relationship with their original designer (and the source files). They’ve then had to pay a new designer to recreate it, which is such a waste of time and money.
Do yourself a favor and specify that you want this variety of file types at the beginning of a project — don’t settle for just a JPEG.