It's similar to electric charge. A particle can have either a positive or a negative electric charge, and if you combine two particles, one positive and one negative, the combination has no electric charge and is neutral. Color charge is conceptually similar, but more complicated. Instead of just positive and negative charges, there are really six types of color charge, red, anti-red, blue, anti-blue, green and anti-green. If you combine a color and an anti-color (say red and anti-red) the combination has no color charge. We say it is colorless, color-neutral, or 'white'. Also, if you combine all three (red, green and blue, or anti-red, anti-green and anti-blue), that combination is also colorless. All quarks carry color charge, and can come in all three colors. So an up quark can be either red, blue, or green, and an anti-up can be either anti-red, anti-blue, or anti-green. The strong force is so strong, that individually colored particles cannot be separated, so all strong interacting particles that we detect (called hadrons) are always colorless. There are two main types of hadrons. The first are mesons, which consist of one colored quark (red green or blue) and one colored anti-quark (anti-red, anti-blue, or anti-green), so that the combination is colorless. The second type are called baryons, which consist of three quarks, one of each color, so that they are also colorless.
All quarks carry color charge. Leptons do not carry color charge.