Is king and pawn against king insufficient material?

No. A pawn can promote, and a queen mates easily, so the game goes on. It may still be a theoretical draw if the defending king gets in front of the pawn, but that is a matter of technique, not of material. You have to play it out or agree.

Is a king and two bishops enough to mate?

Yes, if the bishops run on different colours, which they do unless one of them is a promoted pawn. Two bishops force mate against a bare king in under twenty moves. Two bishops that share a colour cannot mate at all, which is why the same-coloured bishop ending is drawn on sight.

What happens online when someone runs out of time with only a king?

The game is scored as a draw, the same as over the board, because the opponent cannot deliver mate. Servers apply the test automatically, so you will see the result appear as a draw rather than a win on time. Keep an eye on it in blitz: trading down to a bare king is a real way to save half a point.

Insufficient material: when nobody can mate

The last pawn comes off, you look down at a king and a knight against a bare king, and there is nothing left to play for. Not hard to win. Impossible to win, by any sequence of moves either player could invent.

That is what insufficient material means, and there are exactly four combinations of it:

King against king.
King and bishop against king.
King and knight against king.
King and bishop against king and bishop, with both bishops on the same colour.

Each of those is drawn the instant it appears, under Article 5.2.2 of the FIDE Laws. No claim, no handshake, no arbiter. The clocks stop mattering.

Why same-coloured bishops cannot mate

The first three are obvious once you try them: chase all you like, there is no arrangement that covers the enemy king’s escape squares. The fourth takes ten seconds of arithmetic.

Both bishops run on dark squares (c3 and f6 even stand on the same diagonal). Whoever takes whom, no mate exists for either side, so the game is over here.

To be mated, a king has to be in check with every neighbouring square covered. A bishop only ever touches one colour. So the checking piece must be the bishop, the mated king must be standing on a square of the bishop’s colour, and the four neighbouring squares of the other colour have to be covered by something else. The only something else is the enemy king, which reaches at most three of them, and the defender’s own bishop cannot help because it is stuck on the wrong colour too. On the edge the count gets closer, and it still never works.

Two knights: the awkward one

Two knights against a bare king cannot force mate. Everybody knows that, and there is a page on why two knights cannot force mate. But watch what happens if the defender helps.

White plays 1.Ng6 mate: the knight on g6 gives check, the other knight covers g8, and the king on h6 takes g7 and h7. Black had to walk into the corner to allow it, but the position is legal, so the mate exists.

Because that mate exists, two knights are not insufficient material. The rules ask whether a mate is possible by any series of legal moves, not whether you can force one against a defender who is trying. So a flag fall in that ending is a loss, not a draw. It is one of the few places where the Laws and common sense pull in different directions, and the Laws win.

When a flag falls

Article 6.9 is the one to know. Run out of time and you lose, except when the position is such that your opponent cannot checkmate you by any possible series of legal moves. Then it is a draw.

So: your flag falls and your opponent has a king and a bishop, draw. A king and a knight, draw. A bare king, draw, however much time is on his clock. A king and two knights, and you lose, because of the position above. There is more on the clock side of it in what happens when your time runs out.

This is worth playing for in a lost blitz game. If you are getting mated but you are also winning material, trading everything down to a bare king turns a loss into a half point the moment your opponent’s technique wobbles. The related trick, holding a position nothing can break into, lives on dead position, and the drawing resources themselves are drilled in puzzles where equality is the whole point.