Oberyn Martell was the younger brother of Doran Martell, the ruler of Dorne. As he was the 2nd son, he was given more of a warrior's life. I think most fans liked him as his fiery temper, skill with a spear, and his interactions with almost everyone in King's Landing never really was a dull affair. He even makes Tyrion Lannister lose most of his words, as we clearly see his passion to avenge his sister Elia and her Targaryen children live deeply over a decade later.
Probably the most proud Lannister-hater ever, Oberyn strangely made more cordial talks with Tywin Lannister and Queen-regent Cersei than you'd expect. He accepts a seat on the Small Council but clearly shows no respect for the office like not standing for the Hand of the King. He also doesn't have any real power there anyway, while he's on the council he is not a Master of Laws or Ships Or Coin or whatever.
Tywin allows him to judge Tyrion for the death of King Joffrey hoping the Prince of Dorne would vote guilty. Though in trail by faith, it's a 2/3 decision and with a puppet like Mace Tyrell, he wasn't really needed. He would of probably voted for Tyrion's innocence I think if Tyrion had not demanded a trail by combat.
Speaking of the trail by combat, the Red Viper eagerly accepts the challenge to kill Ser Gregor Clegane (The Mountain) in one of the best battle scenes in the whole series. Too bad he was too worried about mocking his sister's killer to keep his guard up.