Here are some guidelines:
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$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . , . . . 2 . X . . . |
$$ | . . O X . X . . . . . . . . . . . . . |
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- Click Here To Show Diagram Code
[go]$$W
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$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . , . . . 2 . X . . . |
$$ | . . O X . X . . . . . . . . . . . . . |
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With the wall in place, when Black plays

, he not only threatens to take a large chunk of territory at the bottom, but also greatly increases the value of an approach at
a. White will be hard-pressed to invade the bottom successfully without giving Black too many points along the right edge in the process as he's being chased out.
Black can do this because his wall:
a) sketches out a territorial framework;
b) is strong and difficult to attack;
c) is a wall, and White doesn't want to run smack into it.
So the wall, for the moment, is strong. Careless play can change this, and play must take place accordingly.
'Influence', to me, is linked to the function of stones. The wall in the above diagram is almost useless if White has a live group right next to it. At that point, it has lost its influence. The reason it has influence is because of the potential damage in can cause White. I view influence as damage a stone can do if used the right way. Back to the diagram!
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$$ | . . . O . . . . . 1 . . . 4 . O . . . |
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$$ | . . . . . . . . . . . . . . . 5 . . . |
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$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . , . . . . . X . . . |
$$ | . . O X . X . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
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- Click Here To Show Diagram Code
[go]$$W
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$$ | . . . . . . . . . . . . . 2 3 . . . . |
$$ | . . . O . . . . . 1 . . . 4 . O . . . |
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$$ | . . . . . . . . . . . . . . . 5 . . . |
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$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . , . . . . . X . . . |
$$ | . . O X . X . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
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$$ +---------------------------------------+[/go]
One feels the stone

has influence. Part of the reason is if Black approaches on this stone's side, that stone will be placed in a good spot to constrict and attack Black's group.
Stones that have influence are stones that can be a pain if your opponent tries to keep you from getting what you're after. Essentially, skillful play near your own influence stones makes the sequence look like you got
two moves in a row in the process.
Last but not least in our little triad is thickness. Thickness is when you don't have to worry about a group because it is strong (absolute or relative strength). You can then carry out your plans while disregarding that group or, better yet, drive your opponent's weak stones towards it. Again, the idea behind a strong group or wall is that if your opponent has the misfortune of being driven towards it, it's like you have a Go superpower that lets you play a dozen mediocre moves at once, since the wall "suddenly" becomes a problem in close proximity.
All in all, the proof, the onus, lies on the shoulders of the person who has invested in thickness, influence and strength. The other player almost certainly has made some solid gains already, and so play must go on in a way that demonstrates the value of your investments. I personally prefer to be on the influence side of an influence-versus-territory battle. Using these things properly is something that you acquire as you try it out, and perhaps study games of pros or strong amateurs.
And as EdLee stated, it's an investment. And you don't make investments blindly, and have to plan ahead before you make such influence. On the other hand, it is somewhat like magic (even though there is a mechanical reason for its effectiveness). In that sense, you can trust that all things being equal, a strong wall, unenclosed group, or loose framework of stones,
will pay dividends.
So keep using walls, frameworks and remember to make your groups strong! There lies the path to improvement.
