When something happens constantly, it never stops or changes. If you sister constantly complains that she's hungry, she is always insisting that she's starving.
If your dog barks constantly when he's in the car, he doesn't stop until you reach your destination, and if you're constantly thinking about your upcoming trip to France, you can't ever get it out of your head. The adverb constantly comes from constant, or "continual," which is rooted in the Latin constantem, "standing firm, stable, steadfast, or faithful."