Most basic first - this rule is the reason that King vs King is an immediate draw. Neither side has a piece to check with, let alone checkmate with. A position that is a draw because neither side can win is called a "dead position".
Playing against a bare king, a bishop or a knight is insufficient to checkmate with, and therefore K+B v K and K+N v K is always a dead position. Most online playing sites end there, and consider everything else winnable.
K+Q v K, K+R v K, K+B+B v K, K+B+N v K are all endings in which white can even force mate (see an overview of basic checkmates on Wikipedia). Because you ask for it explicitly, here is an example of a mate with N and B:
[FEN "k7/8/NKB5/8/8/8/8/8 b - - 0 1"]
There are some special cases. K+N+N v K is tricky -- white can't force mate, but mate is still possible:
[FEN "6k1/8/6K1/4N3/4N3/8/8/8 w - - 0 1"]
White plays 1.Nf6+, and black avoids mate by going to f8. But if he goes into the corner with 1...Kh8??, then 2.Nf7# mates.
So K+N+N v K is basically always given an immediate draw by the player with the knights in a slow game, because he knows he won't win. It will be awarded a draw by the arbiter if black claims one based on rule G.6 [the old "10.2", before july 1 2014], if the other requirements of that rule are met, and it's trivial to make 50 moves without losing. But if black runs out of time, black loses. And in a blitz game where nobody is writing down moves to count them, that may happen.
K+B v K+B is also tricky:
[FEN "5B2/8/8/8/8/7K/8/6bk w - - 0 1"]
With the bishops on the same color, mate isn't possible. It's an immediate draw.
[FEN "8/5B2/8/8/8/7K/8/6bk w - - 0 1"]
But with opposite colored bishops it is: 1.Bd5# mates.
In most other endgames it is possible to think of a way that one or both sides can be mated. In particular, if there are pawns, it is usually possible that one of them promotes and checkmates later. but there are exceptions:
[FEN "4k3/8/8/p1p1p1p1/P1P1P1P1/8/8/4K3 w - - 0 1"]
To a human it's immediately obvious that the kings will never be able to cross to the other side, so this is a dead position. Engines have no clue, though. Mine thinks white is minutely worse.
And in the recent question about sequences of forced moves, limulus proposed this position with an "infinite loop" of forced moves:
[FEN "8/6p1/1p3pPk/1P3Pp1/1Pp3p1/KpP3P1/1P6/8 - - - 0 0 "]
Neither king can break out of its jail -- so this is just another example of a dead position. Immediate draw.