If you had to show someone the gist and essence of the Columbo character...but only had ONE scene in which to do it....could you pick a scene??
After watching the scene the "student" would have to answer an "explain/describe the character" type question
in other words which scene is quintessential??
If this has been asked before...I apologize...

There are two practically opposite ones I can name.
First, the ending of ANY OLD PORT IN A STORM, because of that "warm moment" between him and Adrian.
Then, the ending of RANSOM FOR A DEAD MAN. Even though he and Leslie DON'T really like each other, she can't resist giving him that little compliment.
So those two endings work in opposite ways.

"You're a professional in your career . . . and so am I"

Columbo to Abigail Mitchell in Try & Catch Me. A very touching scene between two old-timers at their craft.

I like the scene in "A Friend In Need" where Columbo tells his boss: "You must get a lot of those", meaning gut feelings. He says this because the Commissioner called Columbo, a homicide detective, for what at that time should only have expected to be a burglary. Columbo goes on to poke all kinds of holes in the Commissioner's cover story such as why there were no fingerprints on the phone:

Columbo: She didn't answer her boyfriend' phone call.
Commissioner: So she blew him off.
Columbo: Yeah, but she answered the phone at 9:30, so why not also answer it at. 8:30? But DID she answer it? Because her fingerprints were not on the telephone!

You can see the veins pulse in the Commissioner's neck. He is so palpably frustrated that this little insignificant nothing is fouling up his plans. He starts to lose it and then recovers.

To me this scene has great clues and great acting. It shows that Columbo is not intimidated by the powerful, and that he is far more intelligent than he appears.

I also nominate for honorable mention the scene in "Now You See Him" where Columbo first walks on the scene. In his new (uncomfortable) raincoat he quickly dismisses the guests, somewhat to the Sgt's embarrassment; Then he proceeds to breakdown how Jesse Jerome could have fallen in the way he was found.

Also there is the scene in "A Stitch In Time" where Dr. Mayfield resets his desk clock as he learns of his nurse's murder. It's a short scene but when Columbo says what a terrific surgeon Mayfield must be we know that Columbo already suspects him.