What are airline "flag" and "supplemental" operations?
Part 121 certificated carriers are given specific permission within their Operations Specifications (OpSpecs) to conduct one or more of those three types of operations -- Domestic, Flag, and Supplemental.

The OP asked: how are these different than "normal" Part 121 operations. The answer is: they're not. A "normal" 121 operation must be operated under one of those three types, with slightly different rule requirements for each.

Each is defined in Part 110.2. (the previous reference to 119.3 is out of date) The gist of it is as Shawn said: US-to-US location is a domestic operation, US-to-nonUS location is a flag operation, and all-cargo or "large aircraft charters" are a supplemental operation. These are not exact, and there are exceptions and legalese within the 110.2 page-long definitions for each one. Please read the FAR for the complete definition.   

Each individual flight needs to be released and operated according to the rules of the operation type that is applicable to that specific flight. 

Airline themselves are not "domestic" or "flag" or "supplemental" (per definition), but it is common to call an airline a "domestic airline" who only has OpsSpecs approval to conduct domestic operations. But that is not a legal term.  Most (I suspect all?) of the majors are OpsSpec'd to conduct all three types of operations. 

Let's take an example of Alaska Airlines. An aircraft is scheduled for (important) and operated on a flight from Seattle to LAX. This flight meets the definition of a domestic operation, so all of the "domestic" rules would apply to this flight. That same aircraft is then operated on a flight to Mexico City. This would be a "flag" operation, and so some slightly different rules would apply to how this flight is dispatched, released, and operated. (alternate airport requirements, fuel requirements, and how much crew rest was required prior to the flight, etc. are slightly different)

Finally, the Washington State Huskys football team charters an Alaska Airlines aircraft to transport them to the Rose Bowl. The flight from Seattle to Los Angeles would be operated under "supplemental" rules. The previous paragraph's example domestic flight might have used the same exact routing, the same exact crew, and the same exact aircraft. But the rules for this specific flight were slightly different, due to the different type of operation being conducted.  

The air carrier, "Alaska Airlines", is not a "Domestic Airline", nor a "Flag Airline", nor a "Supplemental Airline"... rather, the company's certificate grants them permission to conduct all three types of operation.