This test runs from Friday, September 21 through Sunday, September 23. It will run from 5PM to 9PM (PT) each day. It's available to download right now through the Nintendo eShop.
This Network Test will allow players to run around in the Undead Parish area in both single player and multiplayer. The full game will be released on October 19, 2018. I feel like I had to specify the "2018" part there because of how delayed this release has been.
Do take note that Dark Souls: Remastered is not the same as the Dark Souls Trilogy, which isn't even coming out for the Nintendo Switch.
