I was asked this privately, so I thought I'd explain.
A swarm is not an arbitrary number of bees, it's nothing to do with bumblebees (they don't swarm), and it's nothing to do with wasps (they don't swarm either), and if you had a swarm in your garden, you'd know it, because there's nothing remotely like a swarm of bees in the entire natural world.
Swarming is a behaviour that's unique to honeybees because of their unique colony lifecycle. Bumblebees and wasps have an annual colony lifecycle - that is, the colony dies out in the autumn, after which a batch of new queens and drones (males) have been produced, and the new queens mate, and go into hibernation for the winter, allowing the number of colonies to (potentially) increase in the following Spring. This colony pattern is common to all eusocial bees and wasps... Except honeybees.
Honeybees have a perennial colony cycle, meaning the same colony lives through the winter into the following Spring, without hibernating. This is, essentially, why honeybees produce a surplus of honey when other species don't (bumblebees create little pots of honey, but nothing substantial; it's for their immediate food requirements). This perennial colony cycle means that the same queen survives from one year into the next, indeed, queen honeybees can survive for up to 5 years in some circumstances, although this is very rare, for various reasons, and the mean lifespan for a queen honeybee would, as an estimate, be between 18 months and 2 years.
A queen honeybee produces a set of pheromones which maintain the colony's sense of balance (in fact, the absence of the queen, and thus, these pheromones, can alert the colony to the loss of the queen in less than 45 minutes, and they will begin to prepare a new queen from the eggs she laid before she was lost). As she ages, these pheromones weaken, and when she is around a year old, her pheromones will be roughly half as strong as when she was born, and if the colony is large, the pheromone will be spread out among even more workers. This triggers a biological reaction in the workers, that leads to a change in behaviour.
They prepare to swarm.
The swarm is a reproductive behaviour more or less unique to honeybees. Since queen honeybees cannot survive or establish a nest by themselves, as queen wasps and bumblebees can (queen honeybees are adapted to producing copious amounts of eggs and pheromones, but pretty much nothing else), the colony cannot simply produce new queens and let them fly off to found their own colonies.
When the bees decide to swarm, they will build special queen cells around the outside of the brood nest, and the queen will lay an egg in each one; these eggs are to be raised into new queens. She may lay as many as 100 eggs in queen cells in the space of a couple of days, while the workers reduce her feed to slim her down for flight.
These eggs will hatch into larvae, and then after 8 days these queen larvae are sealed to pupate. As long as the weather conditions allow it, the swarm will now leave the hive. Between 40-50% of the workers, as well as the incumbent queen, leave the colony as one; a swarm. They will then form a "beard" on a nearby object, such as a tree branch, while the scout bees (a subset of the foraging bees) seek out a new home, which is usually a cavity about the size of a beehive. Once the bees have decided on a new home, the swarm leaves again as one, until they reach their destination.
Once they start to establish themselves in their new home, they will build wax comb, and start to store pollen and nectar (honey) on it, and the queen will begin to lay eggs, and voila, a new colony is born.
Meanwhile, back in the old colony, the new queens will start to emerge. Depending on the number emerging, and the number of workers left in the colony, the young virgin queens may leave with more workers, in a smaller "cast swarm", or they may fight to the death until only one queen remains. This queen then leaves the hive between 10-30 times to mate with drones, before she starts laying eggs to continue the original colony.
What does a swarm look like?
Swarms are unmistakable, and as I said at the start of this article, there is nothing quite like a swarm of honeybees in the natural world. You will know if you are looking at a swarm.
I think I have a swarm. Should I get a pest controller/exterminator?
No! Absolutely not. A beekeeper will have the knowledge, experience and confidence to non-destructively remove a swarm except in extreme circumstances where the bees aren't readily accessible. Moreover, most beekeepers will be ecstatic to take a swarm to expand their own bee stocks, so as well as being allowed to live, the bees get a new home with a caring steward to look after them.
It's also worth noting that, despite popular belief, bees ARE NOT generally dangerous while swarming. They will still sting if provoked, but it is actually more difficult to provoke them while they are swarming than in normal circumstances, as they have no brood or nest to protect.
You can find local beekeepers and swarm collectors here.