Text mode art is pixel art that is made out of preset tiles arranged on a grid. It's a bit like a cousin of ASCII art, which you might be more familiar with. :) It comes from the text-only display modes on old computers. I think it's a really interesting art style and it's a lot of fun to make pictures with it, even if it can be REALLY difficult sometimes.
If you want to make text mode art of your own then I would use Playscii or lvllvl. Both of them have their quirks so it's down to which one you like more.
You can find lots of tilesets online but my favorites are MRMOTEXT and Microbe 2.
I get all of my color palettes from Lospec's database.