Uh oh, the hand is going to the back pocket, is he really...
No way!
While ejections exist in the traditional American sports, their versions have nothing quite like the impact of soccer's red card, because they all allow you to replace the ejected player. That means that despite the loss of the player, the team can still carry on with standard play. If the player lost was a star, they might be less effective, but the way the game is played isn't radically changed. The closest the non-soccer fan is going to come to the dynamics of the red card is the power play in hockey, but that expires and the teams are soon on level terms again.
So, what to do? There are at least three levels that have to be addressed when responding to a sending off. From bottom to top these are the personnel, the formation, and tactical plan. And there's one big factor that's going to drive what a coach does to each of these: what kind of player was sent off?
Keeper
Losing a keeper is usually a major blow to a team's chances because there's often a large dropoff between the first and second string. However, beyond that there's the simple fact that keeper is the one truly non-optional position. That's why when formations are listed, the numbers only add up to 10. That there will be a keeper is assumed and it's always a given where he'll be. You can play around with the numbers and places when it comes to defenders, midfielders, or forwards, but there's always has to be one and only one keeper in one and only one spot.
This means that one way or another, a personnel change will happen and an outfield position is going away. If you have a substitution, then it'll be used to get your backup keeper on. If not, someone in shorts is going to have to pull the gloves on. Generally speaking, the most likely players to come off are attacking players. With a less experienced or completely novice keeper back there, the first thought is going to be defensive solidity, and that means the the players most likely to get the hook are the offensive types, the attacking midfielders, wingers, or forwards. In the same vein, the formation and tactical plan would most likely become more defensive in an effort to protect the newly introduced keeper.
Everybody else
I was going to go forward line by line but I realized that there are no guarantees with the outfield positions. It's all much more fluid. Because of the defensive concerns playing a man down will tend to create, the general principle is the further back the field the player is, the more likely changes will be. However, this will depend greatly on the previous preparations the team has made. For example, losing a defender is much more likely to result in a substitution than losing a forward, but if the personnel on the field have a lot of training playing in a formation with fewer defenders, say a 3-4-3 or a 3-5-2 instead of a 4-4-2, it might not be so hard to adapt to a 3-4-2 or a 3-5-1. Further, the tactical plan might well become more defensive, but it might not if the personnel on the field are capable of maintaining attacking pressure without leaving dangerous gaps in the back.