If you have to changes planes on the way to your final destination, checking through your baggage means the airline takes care of transporting your baggage from one plane to the other. If you pick it up, you get it and check it in at the next flight. For example, if I fly from Philadelphia to Dallas, I might have to change planes in Atlanta. If I check my bags through, I check them in in Philadelphia and pick them up when I get to Dallas. If not, I have to get them in Atlanta and check them in again for my flight from Atlanta to Dallas.

Check through suggests you have more than one flight and your baggage with go from one plane to the other without you collecting it. You pick up the baggage at the end of your last flight.