I have a few thoughts for you after reviewing your recent game. Please ask any questions, as I would like to be as clear as possible.
1) Cutting Stones and Counting Liberties, Part 1: Winning the Fight you can WinThere are going to be fights. Lots of them. In close combat, each player's troops are cut off from each other by the opponent's troops, unless one side captures a group. Groups need liberties to survive, and a group with more liberties is better off than a group with less.
I think something that will help you immensely in your games will be to count liberties in local fights. A position from your game:
$$Wcm21 Black to move
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O a . |
$$ . . . . . . B O # b |
$$ . . . . . 1 W # # c |
$$ . . . . . . X @ @ x |
$$ . . . . . . X z y . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |
- Click Here To Show Diagram Code
[go]$$Wcm21 Black to move
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O a . |
$$ . . . . . . B O # b |
$$ . . . . . 1 W # # c |
$$ . . . . . . X @ @ x |
$$ . . . . . . X z y . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |[/go]
The marked stones are all behind enemy lines at the moment. The

group is in behind enemy lines, but so is the

group. If either of these groups die, though, the survivor becomes safe. The liberties of each are marked. Each one has 3 liberties: a, b, c for

and x, y, z for

. In addition, these two groups are against the edge of the board, with nowhere to run. Either player can win the race to capture by removing the first on his enemy's liberties.
The

stone is also behind enemy lines ... but it only has two liberties. The group that is cutting it off from its allies is the group with

and

. The White group is already ahead with 3 liberties. In addition, these two groups are facing towards the center, so it's easier to gain extra liberties. It's more likely that Black will remain behind in liberties than that black will capture the

and

group.
In the game, black chose to try and fight with the

group, running to the center and looking to capture the

group. That's a lot harder than capturing the

group. This is better, in my opinion:
$$Wcm21
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O . . |
$$ . . . . . 3 B O # . |
$$ . . . . . 1 W # # . |
$$ . . . . . . X @ @ . |
$$ . . . . . . X . 2 . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |
- Click Here To Show Diagram Code
[go]$$Wcm21
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O . . |
$$ . . . . . 3 B O # . |
$$ . . . . . 1 W # # . |
$$ . . . . . . X @ @ . |
$$ . . . . . . X . 2 . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |[/go]
Black gets ahead in the liberty count on the edge of the board, and White has no way to catch up. Black loses the

stone, but the rest of his stones are out of danger with a single move. To be safe, White can play

to remove any chance of the

stone running to the center, so that

is completely safe.
2) Cutting Stones and Counting Liberties, Part 2: Staying Ahead in the Liberty CountBlack instead chose to fight the harder battle, and wanted to pull the

stone out of the fire. This should have been a mistake, as he's already behind in liberties. However, White makes a mistake.
$$Bcm22 White to move
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O . . |
$$ . . . a 3 1 B O # . |
$$ . . . . 2 W W # # . |
$$ . . . . . . X @ @ . |
$$ . . . . . . X . . . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |
- Click Here To Show Diagram Code
[go]$$Bcm22 White to move
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O . . |
$$ . . . a 3 1 B O # . |
$$ . . . . 2 W W # # . |
$$ . . . . . . X @ @ . |
$$ . . . . . . X . . . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |[/go]
In the above diagram, Black has been pushing towards the center in an attempt to gain more liberties. After

and

, Black's group has 4 liberties. White's group has responded to the first push with

and now needs to respond to the 2nd push. White has 3 liberties at the moment, and is behind. If White tries to remove a black liberty (as you tried by moving at a for

), the count becomes a tie at 3 liberties each ... and Black gets the next move. White either has to keep adding liberties or find another way to make his group safe.
Wait a minute ... The

group has 3 liberties, right? There's another liberty race happening against the edge, where the liberty count is 3 as well. Looking at the position, the

group could also be captured to make the

group safe. Since it's White's turn, White can win that liberty race!
$$Bcm22 W S10!
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O 4 . |
$$ . . . . 3 1 B O # . |
$$ . . . . 2 W W # # . |
$$ . . . . . . X @ @ . |
$$ . . . . . . X . . . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |
- Click Here To Show Diagram Code
[go]$$Bcm22 W S10!
$$ , . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . X X |
$$ . . . . . . X X O O |
$$ . . . . . . . O . . |
$$ . . . . . . O . O . |
$$ , . . . . . , O 4 . |
$$ . . . . 3 1 B O # . |
$$ . . . . 2 W W # # . |
$$ . . . . . . X @ @ . |
$$ . . . . . . X . . . |
$$ . . . . . . . . . . |
$$ , . . . . . X . . . |[/go]
Now, even if Black threatens the

group, it has more liberties than the

group, and will be saved.
3) Can't Cut Me!Knowing how many liberties are available for each group is important, but let's back up a second ... did we need to get into this fight in the first place? What happened to make such a mess?
$$Wcm17 Asking for the cut
$$ . . . X . . . |
$$ . . . . . . . |
$$ . . . . . X X |
$$ . . . X X O O |
$$ . . . . O . . |
$$ . . . O . O . |
$$ . . . , . . . |
$$ . . . . O # . |
$$ . . . O # # . |
$$ . . . a O 1 . |
$$ . . . Y . . . |
$$ . . . . . . . |
$$ . . . X . . . |
- Click Here To Show Diagram Code
[go]$$Wcm17 Asking for the cut
$$ . . . X . . . |
$$ . . . . . . . |
$$ . . . . . X X |
$$ . . . X X O O |
$$ . . . . O . . |
$$ . . . O . O . |
$$ . . . , . . . |
$$ . . . . O # . |
$$ . . . O # # . |
$$ . . . a O 1 . |
$$ . . . Y . . . |
$$ . . . . . . . |
$$ . . . X . . . |[/go]
When White played

, the cut at a was left for Black. When a cut is left open, there's a chance where stones can be reconnected. In this case, Black gets the opportunity to reconnect his

group with the stone at

. Is that what we want?
$$Wcm17
$$ . . . X . . . |
$$ . . . . . . . |
$$ . . . . . X X |
$$ . . . X X O O |
$$ . . . . O . . |
$$ . . . O . O . |
$$ . . . , . . . |
$$ . . . b O # . |
$$ . . . W # # . |
$$ . . . 1 W . . |
$$ . . . Y . . . |
$$ . . . . . . . |
$$ . . . X . . . |
- Click Here To Show Diagram Code
[go]$$Wcm17
$$ . . . X . . . |
$$ . . . . . . . |
$$ . . . . . X X |
$$ . . . X X O O |
$$ . . . . O . . |
$$ . . . O . O . |
$$ . . . , . . . |
$$ . . . b O # . |
$$ . . . W # # . |
$$ . . . 1 W . . |
$$ . . . Y . . . |
$$ . . . . . . . |
$$ . . . X . . . |[/go]
Here's an alternative move to consider. The

group is a solid barrier between the

group and the

stone. With some reading (a skill that always needs more practice,

), White should be able to keep the

group separated and may eventually be able to outright capture the group. White does need to be aware of what Black can do with the cut at b and the atari at c (more reading!).