Why is this not a checkmate? I thought I had the king surrounded

Please help me here, wasn't the king's moves surrounded from all sides?

it is called stalemate

I hate this, just when I thought I understood what checkmate was. I thought it was surrounding the king and the king won't be able to move.

*facepalm* I only just saw it now... So if I had the king in check with one of my pieces, I would have won?
What would you do to achieve checkmate in this case?

Same thing with the King having no escape square AND their King must be in check. That is checkmate. Think of it this way:

Checkmate you are attacking the King (check) and they have no place to run, so they are checkmated. 

If the King is not in check, then it is just "mate" - not "checkmate" - the "mate" is "stalemate" and that is a draw. 

I just came up with this now, but yeah, you need to be checking the King to win as well. 

Don't worry; you'll get the hang of it soon enough. Chess isn't that difficult, but it takes some getting used to the rules and then you'll know them with barely any thinking for it needed.

Ah...stalemate. It’s very similar to checkmate except the king is not in check, and it’s a draw. Keep on practicing and I’m sure you’ll win a lot of games in the future!

Yeah, for you to checkmate, you need to check the king first, and provided that he has no means to get out of check, it's checkmate. In this case black has no legal moves, but his king is not in check, so it is stalemate AKA a draw.

36.Nf6+ Kf8 37.Rh8# ABC...Always Be Checking. 

Instead of taking the black pawn on b3, if you had moved K-f6, it would have been checkmate

Also, another thing you should always look out for in an endgame. If your opponent's king can't move anywhere, he may try to get you to remove any of his remaining pieces or have you block them in order for it to be a stalemate. If that pawn on b3 had been on b7, you could have let it wander down the board while moving your other pieces to check mate him. As long as he has a legal move with the king under quarantine, he has to take it.

*Meant 'N-f6' not 'K-f6' in comment 13

The first part of checkmate is umm.... check. 

U gotta have em in check, that's the important part. It's not gotchasurroundedmate, or allyoursquaresarecutoffmate, it's checkmate.

When ur opponent played the pawn to b3 on move 38 it was a trick. U could've just ignored it and played ur knight to f6 CHECKmate as was mentioned above. Ur opponent saw that the b pawn was all that could move and if u took it... stalemate (also mentioned above).

I hope this helps, good luck.

thank you everyone for your help! I'm learning, slowly but surely

I hope I'll be less of a noob by the end of the year haha

I'm so new to this game. Why is it so hilarious?

It was entertaining. 

Ur gonna be fine. Welcome to chess and umm.. these forums can be hilarious or the complete opposite, as you'll see. 

Anyway, this site has a LOT of good videos and other good training stuff. Just look for stuff at whatever level u think u wanna learn. Do a few tactics each day. Ur gonna have fun improving. Set urself up to play a few slow daily games. That's like correspondence chess where u have days per move. U can improve faster with slow chess because u have time to think