Something that happens continuously doesn't stop. If you enter a dance marathon, dancing continuously for an entire day, it means that you keep dancing — without even one break — for 24 hours.
The Latin root of continuously is continuare, or "join together, or connect."Anything that's done continuously happens without any interruption. When it rains continuously for three days, there isn't a single dry moment during that time, and if your friend laughs continuously for ten minutes, you'll start to worry that there's something seriously wrong with her.