'Carry on' is a common English phrasal verb. It has multiple well-established meanings that are not immediately clear to foreign speakers; .
1. To continue; keep going; return to what you were doing. "Very well, carry on with you work."
2. Manage, conduct, be in charge of. "I cannot carry on a conversation with all that noise."
3. Cause a fuss; to be loud or disagreeable. "I wish you wouldn't carry on so."
4. Especially in the phrase "carry on with"; to date, to engage in romantic relations, to have an affair. "He's carrying on with that Simpson girl."
'Carry on' or 'carry-on' can also be a noun referring to a piece of luggage that a passenger keeps with them, rather than checking or storing in the luggage compartment.