A bishop is stuck on one colour for the whole game. That single fact settles the question: against a bare king, it can never mate, because the king it wants to attack has to be standing on a square the bishop can reach, and every square next to that one is the other colour.
So king and bishop against king is drawn the moment it appears on the board, in the same way as king and knight. Not “probably drawn” or “drawn with careful defence”. There is no legal sequence that ends in mate, so the game is over.
Where a bishop really does mate
Now take the same bishop and give it a king that has boxed itself in.
8 
7 
a 
c Two details make that work, and both are Black’s doing. His king has no square because his own men are on them, and the piece next door is a light-squared bishop, so it can never interpose on a dark diagonal or capture the checking bishop. Swap that bishop for a rook and the mate vanishes: the rook drops back to g7 and blocks.
The other bishop mate you meet in games is the contact one, where the bishop lands right beside the king with something defending it.
8 7 
6 a Both pictures make the same point. The bishop supplies the check; somebody else, quite often the defender himself, supplies the cage.
Keep the pair and the problem disappears
One bishop cannot mate. Two, on opposite colours, mate a bare king in nineteen moves at most, because between them they cover every square on the board. That is the practical lesson buried in this question, and it is why strong players hate parting with one of their bishops in a quiet position: two of them are a mating force, one is scenery.
And when you are deciding what to trade in a long endgame, remember which of your pieces can change colour and which cannot. A bishop never can, so a pawn that promotes on the wrong colour square is sometimes not worth chasing at all. A few bishop endgame puzzles will show you that pattern faster than any explanation.