A follow-up is
- the next alternating play in a sequence; or
- after tenuki, the next play in a local area of the board.
In the second case, it may mean a move that follows another move of the same color after some lapse of time (because the opponent played tenuki). When one speaks of a follow-up play to a joseki, however, it means just that the joseki comes to a natural end; and later either player may wish to return to add a play that wasn't urgent earlier.
This is one of the most common concepts in informed discussion and analysis of particular positions, and is much used on SL.
There are slightly different usages to be found here.
There are pages here such as
These discuss the same matters in other common shapes. These are (all three) volatile situations in which it is clear enough to good players that it is important to continue somehow.
There are also well-known follow-up plays in many standard joseki; here there is the difference that joseki which end do so, commonly, at relatively stable positions rather than volatile ones. That wouldn't be true of all tenuki joseki, though.
These pages deal with openings:
A ko threat is normally a play that would be sente assuming no active ko fight. Therefore in the usual run of play you don't expect to play a follow-up to it. If however your ko threat is ignored, there is scope for unexpected types of follow-ups, which may destroy territory, or lead to the capture of or severe attacks against substantial groups of stones.
In fact, the same can be said for the player winning the ko: the next local play after that may be big. One kind of picnic ko is seen when the follow-up to capturing the ko for one side is 'too hard to handle'.
While a small play with a large enough follow-up is probably sente, and a large play with a small follow-up probably gote, there is a cross-over region where the play and its follow-up are of similar size. This leads to the ambiguous position theoretical concept, at the precise point where there is a difficulty resolving the distinction.