Is it possible to transfer someone's memories and personality to someone else by transplanting a certain brain organ, and if so, what organ - or organs, perhaps - would be able to do this?

The brain itself is an organ.
The frontal lobe is the part mostly responsible for personality.
I know of no way the frontal lobe can be switched from one human to another.

That would be a great story

The story has been done.
Star Trek, Turnabout Intruder.

It won’t be realistic but if you do some good research, it might make a good fantasy or science fiction story.

Maybe if you match, precisely, tbe chemistry from the brain you want to copy, and do a sort of fluid transfusion, while also retrieving a few cells (of the frontal lobe), you can fool those cells and cause them to “remember” in the new brain. Repeating the process, each time putting cells back, and capturing a few new ones.
Very tricky stuff.
Precision is mandatory.
Patience is crucial.

Well I’ve never heard of a brain transplant, so at current tech and understanding, no. It’s not really possible. Realism is fine, but it gets to a point, when writing fiction, where absolute realism hampers the creative work. Pretty much all fiction takes some artistic license and requires a certain degree of suspension of disbelief.

Here is the Star Trek episode on that: https://en.wikipedia.org/wiki/Spock%27s_Brain