When playing in Adventure Mode, you will have the opportunity to loot a Rift Keystone from an elite monster.
These can be used at the Nephalem Obelisk near the Main Camp. Usually you can see it on the mini-map at the town or main camp as a obelisk icon.
A yellow portal opens. Click it and your character will be teleported to a random dungeon.
A progress bar will appear on the right side, just beneath the mini-map. It goes from 0% to 100%. As you kill monsters, that bar increases. At 100% the Rift Guardian will spawn near you.
Below is a video I took of my Crusader level 26 entering a Nephaelm Rift. The video is about an hour long. You can skip everything after I kill the Rift Guardian. I stayed a while to clear all the five dungeon floors. If you see me smashing monsters with Fist of Heavens, I heard a few hours later that the ability is bugged. Supposedly hits 10 times harder than intended. If you are playing beta, I don't recommend exploiting bugs. Might get banned.
The feedback I have heard so far is that Bounties look more appealing and rewarding than a Nephalem Rift. As usual, the developers listen to fan feedback, and chances are there will be significant changes. Remember, this is still Friends & Family beta. Nothing seen in this video is final.