Basically you get another Act - it has 3 bosses - a wad of zones - 10 level increase - unique legendaries to the expansion.
Crusader Class
Adventure Mode - Short fast tasks in each act a total of 6 which can be repeated and are random each time you login to the game.
Example might be Act1 kill Skele king as a task, clear a specific amount of mobs in a specific dungeon as another task, kill a specific rare in another zone as a task, complete a specific event as a task. Doing these nets you exp and gold as a reward, also the zones are as they normally would be with elites and mobs etc, so you get loot from them too.
Complete all 6 and you get a reward from the quest giver npc, this is more exp, more gold and also a cache which has crafting mats and items / plans inside which can even be legendaries i believe. Also you get fragments which leads to the next bit of adventures
Nephalem Rifts - you open these up with a number of fragments (8 i think?) it opens up a portal in town, entering the portal spawns a unique dungeon with a few levels, each level is a different theme to the previous it seems, they can be full of any type of mob, and are usually packed with elite packs also. There is a progress bar that fills as you clear the dungeons mobs, once it hits 100% a boss spawns, killing him results in a tons of loot / gems / crafting mats etc and also blood shards
Blood shards can be used to trade for items with a ? on them, so you dont know the property, its a total gamble, i however got a legendary belt from them but have mostly gotten rubbish.
And thats pretty much it in a nutshell