[alisonatwork]

Outside of VNs (visual novels) and IF (interactive fiction) which has already been mentioned, these sorts of games may also be described as walking sims or narrative adventures nowadays.

Some mainstream studios you might want to check out are Dontnod Entertainment (Life is Strange, Tell Me Why), Red Thread Games (Draugen, Dreamfall Chapters), The Chinese Room (Everybody's Gone to the Rapture), Campo Santo (Firewatch) and Fullbright (Tacoma, Gone Home). There is tons of quirkier experimental stuff on Steam and Itch too. Note that some walking sims are more about exploring the environment and there isn't much dialog, while other ones feature more conversations. Dontnod tend to be more on the conversation-y end if that's what you're after.

You could also check out point-and-click adventures, which are a direct descendent of IF. They require a bit more puzzle-solving, but the pacing is often very good in modern (post-2005) games and it's a nice way to experience dialog-heavy stories. A good place to start might be older releases from Daedalic Entertainment or Wadjet Eye Games, perhaps.

[JeremyReimer]

These are all great examples. I would add "Adios", which came out recently, is short and bittersweet, and has some of the best writing I've ever seen in a video game.