The knight vs pawn endgame usually is not very difficult.
The pawn will promote if the knight and king are far away.
In this position white wins after 1.a6. Even if it would be Black to move first, there's nothing he can do to stop the pawn from queening.
The pawn will be stopped if the black king is in front of the pawn (just like in regular pawn endings).
Here, if Black is to play, he's able to get his king in front of the pawn. After 1....Ke6; 2. Kb4, Kd6; Nothing can stop the king from reaching c7.
If however White had the first move, White would be winning. After 1.c7 the pawn will promote.
Now, let's bring the knight a little closer.
This position is always a draw, no matter who has the first move.
Can you see why?
After 1.c7 (White's best try), Black covers the queening square with his knight, starting 1....Nc4+; 2.Kb4, Nd6;
When your knight is in front of the pawn, this secures a draw in most cases.
The white king can attack your knight all he wants, but he will never capture it.
You'll just move the knight away, while still protecting the queening square.
Do you see how easy this endgame really is?
Let me now show you the last part, the rooks pawn.
At first sight, White seems to be winning after 1.Kb6.
Black has an interesting drawing mechanism in 1....Nc8+.
The idea is to either give a check, cover the a7-square, or play a double attack.
3. Kc7, Nb5+;
4.Kb6, Nd6! preparing a double attack.
Now 5.a7 doesn't help because of 5....Nc8+, winning the pawn.
So White may try 5. Kc6.
Now Black is able to cover the a7-square again. By now White will discover he can't make any real progress.
5....Nc8; 6.Kc7, Na7;
In this final position, white has no other way to make progress. It's a draw.
You may want to practice this a little, to be sure you're able to defend this position. If you're defending this, you'll only draw this knight vs pawn endgame with accurate play.
The rook's pawn on the seventh rank is the exception to the rule.
In this case the knight can't defend the position on its own.
You probably noticed it already.... In this position 1.Kb7 wins easily.
Now we'll give the knight some help. Let's move the king a little bit closer.
In this position Black is able to draw because of a stalemate. You may have seen the idea before. It's a part of the pawn endings.
1.Kb7, Kd6; 2.Kxa8, Kc7;
In this final position, the black king has an important role. If it's close enough the draw is secured.
I'll give you another position to study. It should be easy to solve now.
Is the idea clear to you?
If White is to play, he'll win starting 1. Kb7. The black king will be too late to deliver a stalemate.
If Black is to play, he'll draw with Ke5!.
This is a well known position from endgame theory. It's called Stamma's Mate.
If White is to move, the king has to play 1. Ka8. After 1...Nc8, White has to play 2.a7, sealing his own fate, due to 2....Nb6#.
When Black has the first move, Stamma's mate follows another path. Now 1....Nb5+; 2.Ka8 (forced), Kc8; 3.a7, Nc7#;