"You need HOW much pipe?!"
To solve this puzzle, you must create a single closed system of pipes that covers every square in the grid and has no loose ends. In other words, if you imagine yourself inside the pipes, it must be possible to travel from any square to any other square.
Ignoring reflections and rotations, there are only four possible types of junctions: those with four pipes, those with three pipes, those with two pipes at a right angle, and those with two pipes in a straight line. We'll call those X, T, L, and I respectively, since they resemble those letters.
Some squares have letters in them; these letters represent the junctions to which they are directly connected. For example, if a square has the letters XX in it, then that square's junction is an L or an I, and it is connected to two X pieces. LITX means the square has an X junction in it, and it is connected to an L, an I, a T, and an X.
Once you've filled in the grid, start at each of the colored junctions and make your way to the upper left corner.