But that pawn was never protecting that space in the first place. That's not one of the squares that is weakened.
Space in Chess

I all seriousness, and to the orignial question, I generally think of space as the area behind my pawn chain and as a result, pawn advances, to my mind, are critical in claiming it. Space is just another one of the considerations for me withr respect to the quality of a move, and while it can be offset somewhat by weakened squares left in a pawn's wake, it's usually a trade off I'm willing to make all other things being equal.

Don't make unnecessary pawn moves in the middlegame, thinking (somehow) this will let you "control more space."
On balance, moving pawns (to gain space) because you have (already) developed all your pieces will usually weaken your position.
When in doubt, "make a plan," and maneuver your pieces instead.

You can gain space and weaken squares behind it. You have to assess whether it is worth doing this. You may gain more important squares.
In a way every pawn move except queening weakens the square behind it.
I'm going through a phase where I try to move the queenside pawns as little as possible, except in situations where I obviously have to move them or in specific openings where it's part of my specific repertoire.
+1

And if the OP is really a USCF 1900 player (as per his profile), he already knows these answers. Duh?
I understand the basic concept of space as all the safe squares that your pieces can use, but I still don't know exactly how I should be thinking about it. For instance, some players suggest counting the squares controlled in your opponents territory, and others consider the squares behind your pawn chain. Also, most importantly, do you always consider pawn advances as gaining space even if they weaken key squares like when you advance the center to early?
In order to get a comprehensive perspective of advantages and disadvantages in space in any chess position, you must first consider whether you are analyzing dynamic or static factors of the position. Usually space and time advantages and disadvantages tend to be of a dynamic nature, and must be converted to a material advantage of a static nature. There are dynamic space advantages that can become static in nature and endure to an eventual checkmate. In other words, the restricted space cannot be loosened by any move(s) prior to checkmate being delivered by the player with the space advantage, [ the restricted space does not permit the defender to move a piece or pawn to a certain square that defends against the threat of mate because that square is already occupied by another one of his pieces.] This is the essence (the Achilles Heel) of a static space disadvantage.
Definitions:
Static Factors in Chess- having to do with fixed physical characteristics of the position that usually endure into the endgame (i.e. the actual squares on the board and their relative control by either White or Black)
Dynamic Factors in Chess- having to do with the power of the pawns and pieces and the imbalances of power on contested squares on the chess board (i.e. weak color square complexes). These advantages and disadvantages of a weak color square complex usually have to be converted to dynamic time advantages and disadvantages and eventually result in static material advantages and disadvantages. The main danger with dynamic advantages in space is that they are temporal in nature and with enough counterplay from the opponent have a tendency to dissipate.
A.Static space advantages and disadvantages
1.Static space advantages -- the methodology for exploiting this type of advantage is best described by the checkmate procedure described above
2.Static space disadvantage -- the Achilles Heel described above
B.Dynamic Space advantages and disadvantages -- weak color square complex, binds (i.e.Maroczy Bind pawn structure) A common strategy when one possesses a space advantage is to alternate attacks against the opponent's position. The opponent's diminished capacity to defend due to lack of space produces weaknensess in his position that can be converted eventually into static material advantage(s)
For a comprehensive explanation of the above see "Pawn Power In Chess, by Hans Kmoch. In the book he writes with enlightening and insightful detail about pawns (their rear spans , front spans, and both of their side spans) and the effect on dynamic and static space advantages and disadvantages and many other concepts concerning levers, rams, etc.

Yawn. It's too easy to plagiarize on the Internet. Give it a rest, @Tranny.
And you, @I-Bass-Trick-Off, you should know better.

I all seriousness, and to the orignial question, I generally think of space as the area behind my pawn chain and as a result, pawn advances, to my mind, are critical in claiming it. Space is just another one of the considerations for me withr respect to the quality of a move, and while it can be offset somewhat by weakened squares left in a pawn's wake, it's usually a trade off I'm willing to make all other things being equal.
I think of space in a similar way. Those pawns, backed by pieces, form a territory one can claim - and deprive the opponent from using. Winning chess is all about control - control of squares and space. The more space you own, the more options you have and the less your opponent has. Allowing your opponent too much space is to face a potential avalanche.

Having repeatedly found myself on the wrong side of a fight for space, I'm all too familiar with just how constraining and frustrating a cramped position can be, and how many tempi it can cost to unbind your pieces. It's the kind of thing I delight in doing to my opponents and dread having done to me, so in closed positions in paticular I put quite a premium on space.

Whoa! transpo! Deep, dude.
Grobe, I seem to deal with what you discuss (#28) in most games. In the middlegame, in the back of my mind I'm thinking that if I move that pawn, it can't come back! And I hesitate. Even though I want to gain "space" (squares), I constantly think that once that pawn is pushed, I can't bring it back, so there is a possibility it may hurt me later. Then again, I don't want to be cramped, as you put it, in closed positions.
Just one of the many reasons I stink at this game.

Cramped positions only bother me if I have no clear source of counter-play. There's nothing wrong with a cramped position if you've got a definite goal to play towards. The Scheveningen Sicilian is an example... or many lines in the King's Indian.

Where I get into trouble is moving pawns simply because I don't have a better plan. I still struggle with the ability to identify when quiet waiting moves are called for. Sometimes you don't have a pressing move available and need to turn the move back over to your opponent to let them make the mistake. Usually I just move a pawn and blunder instead....

Space is good as long as it's yours and the other guys is jammed against the far side of the frontier. But once he figures out how to get inside, it doesn't always seem like those advances were such a good idea after all :-(
I understand the basic concept of space as all the safe squares that your pieces can use, but I still don't know exactly how I should be thinking about it. For instance, some players suggest counting the squares controlled in your opponents territory, and others consider the squares behind your pawn chain. Also, most importantly, do you always consider pawn advances as gaining space even if they weaken key squares like when you advance the center to early?