Elves have a very loose naming system, where there are only few hard rules.
Everybody has a forename, given at birth, and nobles have a house name. Thus, Fanriel of House Drangleic. House name can be used as a surname, such as Fanriel Drangleic, but not always, and when they are it usually denotes being either the head of the house or being part of their immediate family. Thus, Tyrion and Teclis are part of House Emeraldsea, but are not called Tyrion Emeraldsea and Teclis Emeraldsea. Fanriel avoids calling herself Fanriel Drangleic due to the complicated and painful situation surrounding her house.
Originally commonborn didn't have surnames, as houses were seen as the privileges of the nobility, but instead have assumed names: epithets either chosen by the elf in question or given to them by their community, describing character traits, physical appearance, or what have you. For an example, Fanriel might be called Fanriel Strongarm, referencing her physique, or Fanriel Wisdomseeker on the account her varied studies.
But assumed names are not exclusive to the commoners: nobles can also have them, like Caledor Dragontamer. And not everybody has one, it's entirely optional and subject to huge amounts of regional cultural variance. Further complicating this is that with the rise of the merchant class, particularly in Lothern, some commonborn families did start to treat assumed names as surnames, passing them onto their children to establish name recognition. And of course, when dealing with humans, many of those elves who don't use an assumed name will often adopt one, again for better name recognition amongst shorter-lived races with their shorter memories, and to facilitate easier communication. And because it makes them look fancy and special in front of the lesser beings. In fact it's not uncommon for elves to have multiple assumed names, used together or differently depending on the context.
That may seem confusing, but it is of course exactly how the elves like it.
Dwarfs, on the other hand, have a much more set-in-stone and formalized naming system. Every dwarf has three names: given name, parental name and clan name, and may have a fourth honour name.
Thus, for an example, the full name of the King of Karak Kadrin is Ungrim Baragrimsson Drakebeard Ironfist. Ungrim is the name his mother gave to him when he was born. Baragrimsson denotes that he is the son of the previous Slayer King, Baragrim. Drakebeard is the name of the Royal Clan of Karak Kadrin. And Ironfist is the honour name that he obtained later in life, referencing his strength in arms and strong will.
Honour names can be chosen by the bearer to signal their intent, such as Thorgrim taking up the name Grudgebearer as part of his vow to right every grudge in the Dammaz Kron and to mark the beginning of the Age of Reckoning, or earned through great deeds, such as the King of Barak Varr becoming Byrrnoth Grundadrakk ("Hammer of Dragons") after hunting down and slaying the Naggarothi Sea Dragon that had eaten his father and brother, or they may be bestowed upon them in recognition of some renowned trait of theirs, such as the Runelord of Karaz-a-Karak being called Kragg the Grim, due to his famously dour nature.
Which name is appropriate to use depends on the situation and the reputation of the dwarf in question, and how they want to present themselves. Given name-parental name tends to be the default and the one that you can always fall back on if you're not sure how to address them, but a dwarf named Thrundin Barunsson who wanted to emphasize that they're part of a famed clan of stonemasons might instead use given name-clan name and go by Thrundin Stonebreaker. But if Barun was a famous (or infamous) dwarf for one reason or another, there might be pushback from his fellow dwarfs who keep calling him Barunsson. Or if Barun was the current Thane of the Stonebreakers, Thrundin might want to go by Barunsson to emphasize his own importance through that way. Ungrim might go by Ungrim Drakebeard when conducting clan business, rather than the matters of the Karak. Runesmiths almost never go by clan name, because it's redundant: all Runesmiths are part of the same clan, clan Thungnisson, so using it is often seen as trying to put yourself above other Runesmiths. Further complicating this is that some clan names are indistinguishable from parental names, due to being named after the clan's founder. For an example, Thorgrim's full name is Thorgrim Ullisson Ulleksson Grudgebearer, because his father was Ulli of clan Ulleksson, so named because they were founded by Thane Ullek during the Golden Age. You are just expected to know these things.
Those who have honour names will almost always use them, but if other dwarfs feel like it is unearned or they're not living up to it, they might show disrespect by refusing to use it, or by giving them a new honour name that they feel better reflects their nature. For an example, Snorri Halfhand, son of Gotrek Starbreaker, got that name because several of his fingers were bitten off by a rat when he was young, and it stuck because he was perceived as rash and reckless by his elders. And naturally, as in all things dwarven, there is a strong element of senior privilege in play: nobody would dare to call Kragg, the oldest dwarf alive, anything other than "The Grim", and he can call whoever he wants whatever he wants, even the High King.
If you've noticed any similarities between the two systems don't worry, they're entirely coincidental.
