It's actually a public test region, not realm, and it's been running in South Korea for a little while already (that's where these Golden Gun images came from). But now it's live in the West as well, and the currently-in-development patch is live on it and waiting for some love. To take part, restart Battle.net, go to the Overwatch tab, select “PTR: Overwatch” from the Region/Account drop-down menu, and click on install. When all is done and updated, click “Play,” and your PTR adventures will begin!
The PTR is only available for the PC, and while accounts for all regions (except China) can take part, the PTR is hosted in North America, so players who live in other parts of the world may experience higher-than-usual latency or other performance issues. A snapshot of your live account will be transferred to the PTR when you make the jump (although it might be slightly out-of-date), but progress made on the PTR will not transfer back.
The PTR is currently hosting the Competitive Mode patch, which makes significant changes to the game including more severe penalties for bad behavior: Quitting early or going AFK will leave you ineligible to join another game until the previous game has ended, and if you keep it up you could wind up removed from Competitive Play entirely. The full Competitive Mode patch notes, as well as more FAQs about the PTR, can be found over at playoverwatch.com.