This is the way I teach it. It works better one-on-one since there are always questions, but FWIW, here goes (it's a copy and paste from a personal FAQ)...
The single most important thing to learn is, there is a difference between the privileges of a commercial "pilot" certificate and the requirements for engaging in a commercial "operation." If you fly for compensation, you are exercising a "commercial pilot" privilege. If you provide both pilot and airplane to transport other people or their property, you are engaged in a "commercial operation."
To give simple examples, if you are hired as a corporate pilot to fly a company airplane or to ferry someone else's airplane, you are only providing them commercial "pilot" services. Their airplane; your services. OTOH, if you take your C182 and start doing air tours or start transporting cargo for someone, you are engaged in a commercial "operation."
Get that difference, and you are way ahead of most in understanding this stuff.
From there, move to FAR 119.1(e). Most commercial "operations" (providing aircraft and pilot) require an operating certificate of some type. Might be Part 135 (charter and some smaller airlines), 121 (the big airlines) but some kind of operating certificate. FAR 119.1(e) is a list of commercial "operations" that do not require an operating certificate. That "air tour?" 119.1(e) says, "no operating certificate required," although there are some other requirements. Transporting that cargo? No 119.1(e) exception, so you need an operating certificate.
There are two other, related, concepts you need to understand. One is the difference between "common" and "private" carriage. "Carriage" just means providing airplane and pilot for transporting persons or property. "Common" means offered to the public, or, as the FAA puts it, a segment of the public. "Private" means only with a very select few. The difference between the two is the other concept: "holding out," which just means "letting people know."
If all that leaves you thinking the lines are blurry, you are right. After all, how does anyone know you are available to transport them unless you let them know? The reality is, it's so blurry, it doesn't make any practical difference. FAR 119.23 says "private" carriage requires an operating certificate, with some exceptions. Bottom line, public or private, if you provide airplane and pilot for a purpose other than one listed in 119.1(e) , you need some kind of operating certificate. But you still have to understand the difference because it gets tested.
I wouldn't worry too much about more than understanding and being able to apply those basics - (1) your privileges are as a commercial pilot; (2) they don't automatically allow you to engage in a commercial operation; and (3) there needs to be some specific regulatory authority to engage in a commercial "operation," such as one of the activities permitted by 119.1(e) or under a Part 135 operating certificate.
Yeah, I know. It's complicated. That's why lawyers get paid big bucks to draft things like leases between a parent company and its 100%-owned subsidiary that does nothing but provide flight services only to its parent designed to prevent the subsidiary from being treated as a Part 135 operation.
Fortunately, it's mostly about understanding the basics such as, "does your commercial certificate allow you to take your neighbor and his family to their vacation destination in an airplane you own or rent and be paid for it? The answer is no since that would be a commercial "operation" and require a Part 135 operating certificate. Truth is, most DPEs don't understand more than the basics either.