The beard is the unsupported part of a head then sweeps downward. Here's an example:
Reason you (may) want one is:
1.) On a fighting axe like a tomahawk or battle axe, you can get a long sweeping edge while saving a lot of weight versus a solid head.
2.) On a working axe, it allows you to choke up with your hand directly behind a good portion of the edge for butchering, fine carving and hewing, giving you better control.
For full-on heavy swings like when felling, bucking or splitting logs, it's not so good, as it is weaker than a solid bit, and the lighter weight, while easier to carry will not bite as deep.