Why is it important to develop your pieces to put "pressure" on opponent's pawn(s)? I hear this a lot from YouTube videos in which experienced chess players commenting as the game develops. It's "just" a pawn! Why design a tactic or strategy around a pawn?
Attacking a piece is often not very useful: the piece does not feel any pressure but just moves away. A pawn won't escape so easily, so attacking a pawn forces the opponent's pieces to defend it and thus limits their freedom. There are a few possible gains of an attack on a pawn: the opponent cannot get enough pieces to defend the pawn so you immediately win the pawn, or his pieces are so badly placed defending the pawn that you can gain advantage on another part of the board, or the pawn has to move and leave weak squares on both sides of it because it cannot ever again protect the squares that it attacked a move ago.
In chess, pawns are the only pieces that cannot move backwards and the ones that have the less mobility. Thus, while in the opening a knight or bishop could simply retreat or be exchanged, a pawn usually has not so many options.
This is specially relevant in the opening. Consider for example the moves:
1.e4 e5 2.Bc4
In this position, black has may good moves:
Nc6 being the best,
d6 also being playable. Now, consider the moves:
1.e4 e5 2.Nf3
e5 pawn is attacked and it black sees his options severely restricted if he wants to defend it, only
d6 being sensible in order to do so.
Another important reason is that sometimes the pieces are unable to defend the pawn. This usually happens in the endgame or late middlegame, where some material has been exchanged, but also in some opening variations. To defend the pawn and not lose it, he has to be advanced, leaving behind him weak squares. Around those weak squares a player may be able later in the game to infiltrate the enemy territory and achieve a winning position.
As a generality, you aren't going to win a good chess match with a singular, huge move that completely devastates your opponent. (If you do, they are not a worthwhile opponent and the playing level in question is novice.) Therefore, you need to coordinate attacks to be synchronized and cumulative. The more advanced the player, the more intricate and multi-faceted the attack. As your skill increases, you will start to use all of these minute details to your advantage, aspiring that they all add up to an insurmountable attack vector.
So, the more advanced you get, the more that these "tiny" things really matter.
First of all, the advantage of winning a pawn is usually enough to win the game (among good players). That's why they're worth attacking.
Second, pawns have the least mobility, that is the least ability to "run away." Their "tininess" also makes them the most vulnerable.
Third, because of one and two above, your opponent will (or at least should) make a major effort to defend pawns. In so doing, your opponent may create other (positional) advantages for you.