Pawns are very interesting -- they are the weakest member of our chess army and at the same time able to promote into the strongest. When we talk about the value of different pieces we usually speak in terms of how many pawns each piece is worth.
Unlike other pieces that are free to roam the board, pawns only move in one direction. It is this limitation that makes it important to understand when to advance a pawn to the next rank -- because there is no going back.
Open and Closed positions
When opposing pawns are locked together face to face the position is considered "closed". Mobility is limited and advantages in development are less important. Knights often do better than bishops in these closed positions.
When pawns are exchanged and the center of the board is not occupied by pawns, the position is open. The player with the lead in development has an advantage. Bishops do better than knights in these positions because they are more mobile.
I believe the term for positions where pawns are not advanced -- and hence we cannot tell whether the position will at some later point be irrevocably closed or open is called "fluid". I think a video author here on chess.com coined this term -- but I don't recall who -- maybe IM Charles Gelofre. Either way, their term has stuck with me and it is important to understand that a fluid position may become closed or open depending on what happens with the pawns.
The pawns in front of your king when you are castled are very important to your defense. It is almost always a bad idea to push these pawns. There are situations and openings that do move these pawns, but be wary.
Fianchetto is where you advance a pawn to make room for a bishop -- and though this is typical in modern openings, be aware that this creates a weakness in your pawn structure near your king. You might accept this weakness as a trade off for attacking the center with your bishop, but be sure to understand modern opening theory before doing this.
Most pawns are assigned a value of 1 -- which makes sense because we are talking about how many pawns a piece is worth. However, center pawns are interesting -- they are actually worth more than one ordinary pawn. Some people value them at as much as two pawns. The numbers aren't that import, but the idea here is that your D and E pawns are more valuable than your other pawns -- whatever number you assign to them. This is because these two pawns create valuable space that allows your pieces greater mobility.
Pawns can control at most 2 squares -- the two they attack. If two pawns are side by side they attack twice as many squares as two pawns where one guards the other.
When one pawn is in front of another pawn, the pawns are said to be doubled. This is a disadvantage because the rear pawn cannot advance without the forward pawn advancing first and a single opposing pawn or piece can stop both pawns from moving. There are rare situations were doubled pawns are not a weakness, but largely, you should avoid creating doubled pawns.
If they do not create new pawn islands and are not doubled backward pawns they may not be horrible. If they help you control the center or open lines of attack, they are fine.
When a series of pawns guard each other, they form a pawn chain. They generally create space at the expense of controlling fewer squares. This space advantage might serve to limit your opponent's ability to develop or simply to make some of their pieces less active. The weakest pawn in the chain is the rear most pawn. Another name for this pawn is the backward pawn.
As a rule of thumb, when you have a four or more pawns in a chain, you should focus your attack in the direction the chain points. I believe this is largely due to the idea that you have the greatest mobility in that direction.
Backward pawns must be defended by pieces rather than pawns and are very vulnerable to pressure. Strategically, this a weakness waiting to be exploited.
If a backward pawn has no opposing pawn in front of it, queens and rooks can easily attack it. The backward pawn generally can't advance because it is reponsible for the defense of another pawn.
The square in front of a backward pawn is generally weak and can become an outpost for enemy pieces. Both sides should try to control the square in front of a backward pawn.
Capturing toward the center
When two pawns could possibly capture a piece on a square you generally want to capture toward the center. This has to do with many factors and varies from situation to situation -- but his is a good rule of thumb. Factors include doubled pawns and pawn islands. If you determine that it would be tactically bad to capture toward the center, then by all means, don't.
At the beginning of the game, you have 8 pawns with no gaps. You could consider this one giant pawn island. If a hole is created where one file is missing a pawn, there would now be two pawn islands. The more islands that exist in your pawn structure the weaker your pawn structure is. However, these open files can also create opportunities for your rooks -- so creating additional islands isn't always bad.
One can have a king-side pawn majority, center pawn majority, and queen-side pawn majority. It is my understanding that a central pawn majority is very important in the opening and a queen-side pawn majority is critical in the end game. Wilhelm Steinitz paid special attention to the queen-side pawn majority in the end game. Positionally, pawn majorities can be required to mount an effective offense.
There is a concept of minority attack, where the pawn minority attacks the majority which can originate from positions like the queen gambit declined. The idea is to improve attacking chances on the side with more space at the expense of a weak pawns.
I don't claim to understand every strategic consideration around pawn majorities, but understand they are important with regards to creating attacking chances and opening lines to the enemy king.
A pawn is "passed" if there are no enemy pawns in front of it. These pawns are valuable and grow in value the closer they get to the last rank where they can be promoted.
The file that the passed pawn is on is considered half open.
Passed pawns are usually best supported by a rook behind it.
It is important to push passed pawns. On the last rank before it is promoted, a pawn is worth as much as a knight because of the threat it poses.
Outside passed pawns are on the on the edge of the board and separated from other pawns by a few files. These can be decisive in an endgame because a king cannot navigate both sides of the board fast enough.
An isolated pawn is one with no friendly pawns on files near it. This pawn creates a weakness. The square in front of an isolated pawn is weak because no opposing pawn can attack there. The pawn itself requires defense from a piece -- and hence reducing the mobility of the piece that has to protect it.
Isolated Queen Pawn ("The Isolani")
This is an isolated pawn on the D file. The isolated queen pawn is an interesting exception to the isolated pawn. In the middle game it can create attacking chances and be a strength, while in the end game, it is usually a weakness like any other isolated pawn. This pawn can create outposts for knights and open diagonals for bishops. There is a video series by IM Daniel Rensch on this subject here on Chess.com.
Pushing connected pawns takes away space from your opponent and can break open defenses. As it is not desirable to push pawns in front of your king, these should be pawns opposite of where you castled.
Whenever you push any pawn, you create weaknesses behind it. Pawns storms can be effective against a Fianchetto position. Usually a pawn storm attacks a king's position -- which works best if the storming side castled queen side. As it is less common for black to castle queen side, it is more likely white that chose to castle queen side. This does not mean that a pawn storm must attack the king side.
Pawn storms are best left until the middle or end game when the queens are off the board. An example of an opening that may lead to a pawn storm is the Svenshnikov variation of the Sicilian defense.
This blog post is geared toward new players. Not that I am qualified to give advice -- I myself am not an advanced player. I merely hope to help struggling beginners.