King and pawn versus king looks trivial but is full of precise technique. Whether it wins depends on king position, not just the extra pawn.
The winning plan is to put your king ahead of its pawn with control of the squares it needs to advance. If the king leads and you can gain the opposition, you usually promote.
For a pawn that hasn't crossed the middle, the key squares are two ranks in front of it. If your king reaches a key square, the pawn promotes regardless of whose move it is.
To see if a lone king can catch a passed pawn, picture the square with one corner on the pawn and one on its promotion square. If the defending king can step into that square on its move, it stops the pawn; otherwise the pawn queens.
a- and h-pawns are the exception — the enemy king can hide in the corner for a draw.