The other two answers are good ones & correct as far as they go, although I don't think either entirely addresses the question as posted regarding when a pilot has to have both ceiling and vis, or vis only, and the differences between ICAO and FAA criteria.
All approaches will have PUBLISHED both a ceiling and a visibility minimum. Whether you're required to have both of those in order to begin the approach is a separate question. (Or, in some cases, you don't have to have either one -- FAR Part 91 for instance. Under those rules, you can start the approach having neither one. Maybe you're just flying the approach in order to practice the missed approach, and you're okay with the fact that you almost certainly won't be able to land out of it.)
For circling approaches, every authority I'm aware of that requires having minimum weather in order to begin the approach, requires that you must have both ceiling and vis. For straight-in approaches, some authorities require that you must have both, and other require only the published visibility. In FAA rules, flying a straight-in "Vis Only" is generally allowed; some operators (such as the Air Force's Air Training Command, now the Air Education and Training Command, at least when I went through pilot training) require that their pilots have both in order to begin the approach. Major airlines typically have approval to fly straight-in approaches "Vis only."
The idea is, don't start an approach unless there's a good probability that you can land out of it. (Part 91, as described above, is less restrictive here.)
My understanding is that ICAO does the same thing, in that under ICAO rules generally, straight-in approaches may be flown "Vis Only" and the ceiling disregarded for exactly the reason that LNafzinger mentions -- the reported ceiling may not be perfectly uniform and you may well be able to see the necessary runway/lights/environment at your Decision Altitude even though you're somewhat above the published ceiling value. Some nations may choose a more limiting rule that requires both ceiling and vis -- I think at least one nation in the Caribbean does so, although I don't remember which.
Okay, follow-up question: why publish a ceiling if operators are free to ignore it? Answer: besides those operators who choose to require it, it's also used when considering the requirements for alternates and when an alternate qualifies. Those rules have conditions like "if the ceiling is lower than XXX above the lowest authorized approach's minimum ceiling and/or the visibility is lower than YYY above the lowest authorized approach's minimum visibility, then..." Rules like that can only work when the ILS and PAR approaches have the published ceiling values as Jonathan Walters posted.