How can I play Day of the Tentacle?

First, you'll want to grab a copy of ScummVM. DOTT (as well as a bunch of other adventure games from that era) all run on a common game engine. ScummVM is a modern implementation of that game engine which has been ported to pretty much every platform under the sun.

Next, you'll need a copy of Day of the Tentacle's data files. (monster.sou, tentacle.000, and tentacle.001) These come from the original game CD, which you'll need to obtain through whatever method you prefer. Once you have them, just click "Add Game" in ScummVM and browse to the directory they're found in.

You can play it on a modern PC and operating system using ScummVM, this works for many, if not most adventures of that time.

If you'd like, DOSBox also runs really well and is crossplatform too. I use it on my Linux machines to play Commander Keen, Dark Forces, Full Throttle, etc.