When printing from your computer to an ordinary printer, multiple heads spray the colors cyan, magenta, yellow and black in one pass of paper to create a multi-color image. When screen printing an image, each color has its own screen and the t-shirt or other garment makes a stop at each station on the press.
The actual “screen” is a frame stretched with mesh. Just as it sounds, “mesh” is a cris-cross pattern of thread. “Mesh count” refers to the number of threads per square inch - this is a similar concept to the dpi or lpi you refer to with an ink jet printer. Different meshes and mesh counts accomplish different things on press.