When you do something continually, you do it over and over again. If you keep checking your email to see if you've gotten an important message, you can say you check it continually.
The adverb continually is good for describing something you do repeatedly, or on a regular basis. Your little brother, for example, might complain continually that he's hungry for ice cream. The word continuously is easy to confuse with continually — when you do something continually, you take breaks in between, but when you do it continuously, you don't stop at all. Both come from a Latin root, continuare, or "join together."