Punch mitts, sometimes referred to as focus mitts, help with improving punching skills in boxing. They also help a boxer to think quickly when reacting to the punches coming at them.
Boxing coaches usually have these among their personal equipment. However, boxers may want to have their own pair as well. If a boxer trains at several different gyms, and punch mitts aren't always available, it would be nice to have a pair in the gear bag. If a boxer wants pad work, and the coach is unable to hold the punch mitts, another boxer with their own punch mitts may be able to hold them for someone else.
Punch mitts are like gloves with targets on them. One person puts them on and directs the other person to throw punches at the targets, which are usually positioned in the middle. The person with the mitts on calls out the punches by name ("left jab, right hook!") or by number ("give me 1-2-3" which translates to "give me left jab, right cross, left hook"). The idea is to simulate sparring or an actual bout, so both persons need to move around the ring as the punches are being thrown. The person holding should keep them close to their body.
When using them, don't just stick to calling punches out by rote. Give the other person combinations to work on. Have them practice slipping, bobbing, and weaving. The person holding the mitts should apply a little resistance back when the person punching makes a connection. Be careful not to hit back way too forcefully because that could cause damage to the hands and wrists.
The person holding them should pay attention to how the other boxer is moving, and be prepared to give feedback. This will help the other boxer when they spar and/or have bouts. It will also help the person holding if they regularly spar with that person themselves. They are at a good advantage to use whatever information they observe to have an advantage the next time they spar with that person!
Working with the punch mitts can be used as an alternative to sparring. Some days, a boxer may not feel like putting on the headgear and the mouthguard. If done properly, using the punch mitts is as much of an intense workout as sparring.