in a nutshell, 119.1 tells you what you can do as a commercial pilot without an operating certificate (flight instruction, pipeline/powerline patrol, limited sightseeing, etc). The "common carriage" thing relates to "holding out" your services. "Holding out" your services basically means you are advertising your services, whether it be by flyer, business card, word of mouth, or even someone elses word of mouth.
AC-120-12A spells it out a little better, "common carriage" is :
"holding out" a willingness to transport persons or property from place to place for compensation. This "holding out" which makes a person a common carrier can be done in many ways and it does not matter how it is done.
You asked what you *can* do with a commercial pilot cert, and that answer is that you may act as PIC of an aircraft for compensation or hire. What you may not do is hold yourself out as a common carrier. You may however hold yourself out for the exceptions in 119.1 (advertising yourself as a flight instructor is ok, etc).
One of the scenarios i used to give students was a guy comes up to you and asks you to take him to XYZ and offers to pay you. This isnt quite as related to the above, since you may not have held yourself out to perform this service, but in this case that you are providing the airplane and flying the customer "on demand", you are in need of an operating certificate to perform this service. (charter/on demand is not in the exceptions in 119.1).
The alternate to this service is a guy with an airplane comes to you and asks wants to hire you to fly his airplane from A to B. This is ok because you are solely being paid to act as pilot in command of an aircraft, not for the airplane, etc.
The area is very gray in a lot of ways, and imho one of the more confusing aspects to the commercial certificate.
I'll fix my post.