Parents might be somewhat long suffering when it comes to their own children, enduring hours of tantrums or arguing with a sibling; however, when dealing with someone else's bratty kids, fuses are significantly shorter. As you may have already realized, for some reason, it is different when the temperamental children aren't yours, and they can grate on your nerves to the point where you want to wipe that smirky grin right off their faces. The main reason for this issue may be partly due to the fact that you feel you have no control over them, but that can change.
Ask the parent of the other child if she wants you to provide punishment for misbehavior. It is better to know up front if the other parent is OK with you fussing at her child. This is only necessary if the other parent is dropping off the child and leaving. Say something like "So if Susie misbehaves, do you typically do timeout, or is there some other way you want me to handle it?" This way, you are not only asking but also telling the other mom that you plan on doing something, but you are giving her the option of what it is.
Take a breath. It is best to calm yourself before dealing with a bratty child, whether he is yours or not. Count to 10 or breathe calmly for a few seconds and assess the situation. You will be better focused and less likely to overreact.
Chastise the child when you witness the bratty behavior. Sometimes fussing that comes from an adult other than the child's parent is taken more seriously, and causes them to behave for at least a little while. You might say, "John, why did you do that? Did you ask first?" If the child's parent is there, you need to be more subtle, unless she gave you free rein. Instead say to her, "It seems your little one needs a bit of guidance, but I don't want to intrude on your turf."
Place the child in time-out if the other parent is okay with you providing that type of punishment. Drop down to the child's eye level and explain to her that how she acted was unacceptable. This way she understands why she is being disciplined. Once time-out is over, reemphasize what the issue was and tell her that you will place her back in time-out if she does it again.
Separate the child from other children and tell him to play by himself for a while. Ask the parent if this is OK with her before doing it. Explain to the child that if he cannot act nice, then he cannot play with the other children. You could say "Now Timmy, you know what you are doing is bothering the others. Since you cannot play nicely with them, then you will have to play over here alone. In a little while, I'll come get you and you can try again."