A pitch and putt golf course is a course with only short holes — holes that typically are all so short as to require literally only pitch shots and putts. A pitch and putt doesn'tto be that short, but when a golf course calls itself a pitch and putt, expect that most, and typically all, of its holes are that short.
Some other names that might be applied to a pitch and putt golf course — but that aren't necessarily the same thing — are approach course, par-3 course and executive course. All of those fall under the catch-all term of "short courses." Alternate spellings are pitch-and-putt, pitch 'n' putt, pitch-n-putt.
And the use of "pitch and putt" doesn't require that you add "golf course" onto the end of the phrase: If you say, "Let's go to the pitch and putt," your fellow golfers will all understand you are talking about a very short golf course.
How short are pitch and putt holes? That's up to the facility, of course, and any facility can call itself a pitch and putt so long as it has short holes on its course. But a short course that is a proper pitch 'n' putt is unlikely to have many holes longer than 100 yards. Some pitch and putt holes might be as short as 35, 45 yards — as we said at the top, holes that literally require only pitches, chips and putts.
A pitch and putt might be just nine holes or the full 18; it might be part of a larger practice facility that also includes a driving range, bunker areas and putting greens, or it might be a standalone golf course.
Pitch and putts can be great places for any golfer who wants to work on his or her short game, particularly wedge play. And pitch and putts are also great places for beginning golfers. But some pitch and putts are part of amusement-type facilities (batting cages, kart tracks, arcade games, putt-putt, etc.), geared not necessarily to golfers but to groups of friends looking for a variety of things to do, parents with small kids, and so on. So if you want to use a pitch and putt to actually work on your golf game, make sure it's part of a golf facility.