Originally, In the Groove (ITG) was a 4-panel dance game created by Roxor Games based on the earlier (and much more popular) Dance Dance Revolution (DDR) game by Konami. Like DDR, arrows rise to the top of the screen, to the beat of the music, until they reach hit targets for left, up, down, and right arrows. The player has to hit the right buttons when it crosses the hit target to score well and prevent failing the song.
ITG ran on a modified StepMania 3.95 engine (an open source clone of DDR) and introduced features such as mines and hands and support for “mod charts”. ITG was sold as a PS2 game, an upgrade kit for existing DDR cabinets, and as an ITG dedicated cabinet (dedicab). Shortly after ITG gained popularity, it got into legal trouble with Konami and it went away in an official capacity.
This video does a great job of summarizing the game.
The term "ITG" originally was used to refer to In The Groove, the game series developed by Roxor. But over time, ITG colloquially changed to mean ''4-panel StepMania played on a dance pad with ITG judge and life values''. So if you're using a 4-panel dance pad, have OutFox, use life and judge 4 values (which you would be by default if you use a theme like Simply Love), then people would say you're playing ITG. This term is often debated and on its face is inaccurate, since you're not playing the original In The Groove game -- but it's just a manner of speech you'll hear online a lot so it's worth explaining.
ITG uses judge 4 and life 4 in StepMania. See Judgements and Timing Windows for more.