There are several things :
- They are typically (but not always) small lenses. Compact, easy to carry.
- They are typically a good focal length either on full frame or crop frame cameras. On crop frame they make good head-and-shoulders or head portrait lenses. On full frame the field of view is wider, and but still offers good portrait lens angles, or street photography angles.
- They are wide aperture lenses, making them useful for low light or shallow depth of field photography - the later being again useful for portraits.
- Many systems have relatively cheap entry level f/1.8 50mm lenses, making them a popular entry into wide aperture photography.
The small size means they're easy to carry around with another lens, making them really handy to keep around for photo opportunities that pop up. Personally I'd be loath to travel without one if at all possible, particularly to a social event.
Nifty-fifties exist for most DSLR and MILC platforms. On the 4/3 and micro 4/3 systems they would be slightly longer for field of view, but still offer a lot of possible options, but fewer social shooting opportunities.