The Purge: Anarchy isn't so much a horror movie as it is an action thriller.
Let me be clear here I have not seen the first Purge movie, and even after seeing and enjoying this one I still don't have much of a urge to. Anarchy takes an interesting approach to the standard trope of taking a horror movie with a sort of attention getting premise and spamming out retreads of the same thing year after year. Only one movie in the Purge is changing the formula and expanding on it's "universe" with larger themes than just the setup and plotlines that span beyond the typical purge night of terror.
Frank Grillo (Warrior, Captain America 2) plays an unnamed character who is setup in a room loading guns with a picture of his young son sitting next to him and a girl telling him not to go. From this you can pretty much interpret his entire motivations, history and future actions he will take. This may sound like a boiler plate setup and it is but the movie doesn't focus on the why but so much as making him an interesting person to watch. Much like say Batman Gurillo is driven to presumably find the person who killed his son and is not afraid to use his skills to do so by any means necessary. But as it would be too easy for Gurillo to just go it alone a few other characters come along to slow him down. The two other main plotlines involve a struggling mother daughter combo who get caught up in a purging and are saved by Gurillo, and a boyfriend/girlfriend pair who get stranded and of course are saved by Gurillo.
The two side pairs of characters are definitely the weakest links of the movie as their stories are standard fare not unlike Grillo's but without the cool or interesting factor. The daughter has the standard sassy attitude who seems wholly unappreciative of everything Grillo has done for them, and the couple were in the middle of breaking up when this happens (can you guess how their arch goes?). But they actually do a fine enough job of putting Grillo in dangerous scenarios.
The best part of the movie is that it is pretty clear from the beginning that Gurillos character is not in any danger of becoming a standard horror movie victim, in fact every time a bad guy comes after them you can't help but get excited because they have no idea what they are getting themselves into. This makes Anarchy a great blend of an action movie but with a horror/thriller edge that give the fights a visceral feel that most other action movies would shy away from.
Beyond the standard story there is a greater set up happening for the obvious sequels that will come out. There is a rebel group who is attempting to bring an end to the Purge day and a government conspiracy type that has a very Hunger Games like feel to it. These greater stories are not necessarily heavily related to the main plot but actually do have a couple pretty cool payoffs in the movie. Probably the best thing you could say about the movie is that it is thrilling, each set piece has an interesting take on the formula, and it is fun to see Frank Grillo take that setup and shoot it in the face.