An afterswarm is a secondary swarm of bees, one that happens after the primary swarm.
When a bee hive gets too crowded the bees start getting ready to leave the hive. The worker bees allow more virgin queens to develop (normally the virgin queens kill each other, so the workers have to keep them separated if they want a good supply). Next, the queen bee will take most of the older workers and drones and set off to start a new hive. This is the prime swarm. Sometimes there will be one or more afterswarms, generally within a couple weeks of the initial swarm, with virgin queens heading out to start their own nests, each taking a sizable group of workers with them.
Afterswarms are common in large, healthy hives, and they are not inherently bad; in fact, they are a good thing, allowing healthy populations of bees to spread quickly across the countryside. But they do mean that there are less bees remaining in the hive, which means a longer time before the hive will produce useful amounts of honey. Beekeepers will sometimes try to prevent afterswarms; the methods for doing this are essentially the same as the methods listed in the swarm control node.
Afterswarms can be problematic for other reasons as well. Afterswarms are often much smaller than the initial swarm, and therefor the new colony formed will be weaker and take longer to get up to full strength. This is less useful for starting a new hive, and it may be better to discourage an afterswarm, resulting in one stronger hive, than to allow it, resulting in two weaker ones.
Afterswarms tend, obviously, to take place once the virgin queens mature, and as multiple virgin queens are raised at once, it is not uncommon for two afterswarms to leave the hive in short succession, within a couple days of each other. This can make it very hard for beekeepers to control hive population. This process is not left up to chance, of course. The worker bees determine how many swarms they want, by preventing each new queen from stinging and killing her developing queen sisters, until the hive population is at a good level. Beekeepers can manipulate this, either by removing queens or by expanding the hive and thereby making room for the current population.