Changing stance is a big part of gameplay. The essential concept to master is using the correct combat position for each situation.
Generally speaking, as you go down the list accuracy will increase and movement is quieter. On the other hand, as you go up the chain of stance movement is quicker and aiming time can be reduced. This is an obvious trade off system that gives you a lot of options on the battlefield.
Ready
Mercs can move quickly but maintain a higher degree of accuracy than running. It is not as quiet as crouching or going prone but it does hamper noise a little. It is the highest active stance that can allow your squad to aim over objects. Unless you need to cover a lot of ground, this should be the default stance to keep your mercs ready to respond.
Run
You merc moves as quickly as possible but also makes the most noise. They are not planning to fire, so any shots are from the hip and significantly less accurate.
However, mercs can fire very quickly because they take very little time to aim. The impact of these effects is different, depending on the weapon being used.
It is foolish to run and attack with a precision weapon like a sniper rifle. However, shotguns and SMGs are a lot better shooting from the hip.
Firing from this position is never recommended; it is basically used to cover large distances quickly.
Crouch
In a crouched position, your accuracy increases and mercs move noticeably slower. The cautious movement substantially reduces the amount of noise you make.
This position can make excellent use of cover because your merc is low enough to get protection behind many objects while having enough height to shoot over them. This stance is probably the most common for firing in urban areas.
Prone
Your merc lies flat on the ground and has the highest accuracy possible. This is the best spot to be in when shooting from a long distance and you have time to line up your shots.
But, unless you have a clear path or elevated position, it can be hard to get line of sight with enemies when you are in this stance. It's a great position for snipers, but not a good idea if you are being charged by a squad of enemies.
Movement in this position is incredibly slow but equally silent. Going prone will give you the stealthy advantage to sneak into an optimal position.