Laura Dern plays the role of Admiral Amilyn Holdo. The character has been relatively absent from promotional materials for The Last Jedi, although most are at least vaguely aware of the character's existence, and that she has purple hair. Beyond that, not much is known, although she did make an appearance in the novel Leia: Princess of Alderaan. A Vanity Fair article sheds a bit of light on the character.
[VERY MILD SPOILERS FOLLOW]
According to the article, Admiral Holdo has a long history with Leia. She is a quirky, off-beat character (the article states that fans compare her to Luna Lovegood from Harry Potter). Her style is very different from Leia. She also clashes with Poe Dameron, but it remains to be seen what her motivations are. Based on the materials available, Vanity Fair sums up Holdo as:
An oddball, controversial boss with adventurous sexual proclivities
Entertainment Weekly has 4 cover variants coming out, one of which is Poe and Holdo together. The cover suggests that Poe may be at odds with his new boss. It is implied that Poe is Leia's choice to lead the Resistance in the future, which may mean a power struggle at the top of the Resistance. This would be a political aspect to the good guys that we never really saw in the OT with the Alliance. If there is any kind of political struggle, hopefully it is executed better than in the PT.