While "king" in Dwarfish simply means "senior mining engineer", the Low King of the Dwarfs does have much more power than that; he is their ultimate court of appeal. There is no notion of royalty or divine right to rule, but the Low King's word on a case or an issue is the last word.
The Low King is elected by senior members of the more powerful mining clans. The "throne" of the Low King is the Scone of Stone, reputedly a dwarf bread artifact dating back to the first Low King, B'hrian Bloodaxe, fifteen hundred years ago. The seat of the Low King is in Bonk, Überwald, amidst a vast network of underground mines. The Low King also handles diplomatic affairs.
Long ago Hugen was Low King, and was responsible for the inventor Morose creating the game Hnaflbaflsniflwhifltafl (Thud) that would teach young dwarfs the virtues of preparedness, strategy, boldness and quick thinking and the vital necessity of not being too clever by half and pissing off your Monarch.
The current Low King is Rhys Rhysson, from a small coal-mining clan in Llamedos. If you are confused as to why it is the 'low' king remember that dwarves view the world as almost upside down therefore making the 'low' king the 'high' king. Therefore, "the world turning upside down" for Dwarfdom at the end of Raising Steam is, as the new Low Monarch reminds their audience, not a bad thing at all.
Low King Rhys adopted a new name at the end of Raising Steam:
At breakfast, Lady Sybil said to her husband, “It sounds quite an adventure, Sam. I hear the Queen has changed her name to Blodwen. It means ‘fair flower’ in Llamedos. Isn’t that nice? I must write to her.” “She’ll like that,” said Vimes, whose wife’s capacity to remain in touch with everybody she had ever met was well known and sometimes quite useful.