Have you ever wondered what a swarm looks like? This short video is a swarm in progress.
Bees swarm because something isn't up to their specs. If the queen runs out of places to lay eggs, the workers feel too crowded, workers run out of places to store honey, the hive keeps getting attacked or robbed, etc.
Generally speaking, a swarm has TWO things occurring at the same time in order to be considered a swarm. First, you will have a large mass of bees on the front of the hive, around the entrance. BUT, this alone does NOT mean they are swarming. A mass of bees on the front could just mean the bees are too hot on the inside of the hive, so they are coming out to cool off. Second, you have a massive amount of bees flying around the hive and in front of the hive that looks like they are just wasting time, not coming and going from the hive gather resources. Bees are not lazy, so if they aren't working, they are doing something important.
A swarm will almost always have BOTH the mass of bees on the front of the hive AND a bunch of bees flying all over and not working.
These masses of bees are just waiting for the queen to come out and then they will all leave together, looking for a new home. If a hive swarms correctly, they should leave behind a new queen (or queen cells) and enough bees/food for the remaining hive to survive. A swarm is just the way the bees naturally do a split, which is what we do manually before they swarm and to reduce a hives population.
Still not confident you know if it's a swarm or not? Easy way to check: open your hive and look at the bottom of your frames. The piece of wood that makes up the bottom of your 4 sided frame will have queen cells hanging off them if they are swarming or will be swarming soon. If you don't have any of these large cells hanging off the very bottom of your frames, then your hive is not swarming.