Assuming that the OP is referring to the US because you have never been outside the US is a poor assumption. But honestly, this whole tipping discussion is WAY off topic, as is the discussion of assuming a US- or non-US viewpoint.

Getting back to the topic at hand, checked baggage is that which you hand over to the person at the ticketing counter to put in the baggage hold of the aircraft. If your airline tells you "one free checked bag", this is what they mean.

Carryon (the bags you take with you onto the aircraft) rules vary by carrier. Most US carriers are pretty consistent (with the exception of Spirit), European airlines tend to be more strict in terms of size, weight, and number of carryon items. Asian airlines (in my experience) are closer to US airlines. Intra-Asia, I've seen some crazy things being carried on.