Standard hydrogen electrode, why hydrogen?
There are two separate issues here. The first one is defining a zero for potential. In theoretical physics, it is the energy of two charges separated by infinity distance. 

In electrochemistry, this is chosen to be the potential of the hydrogen electrode since that is something that can be made reproducibly and hydrogen was the de facto 'zero' for a number of other quantities.

These days, there are much more practical and easy to use reference electrodes like the silver/silver chloride (as chipbuster mentioned) or the saturated calomel or the mercury/mercurous sulfate. Here is a good list of them from a manufacturer of such electrodes: http://www.koslow.com/select_reference_probe 

Full disclosure: I used to work for a company that resells electrodes from Koslow.