In the world of Star Trek, the transporter is used to move people and objects from one place to another, "beaming" them around. Various characters on the show (notably Dr. McCoy and Lieutenant Barclay) hate the transporter. So, yes, the transporter is fictional, but that doesn't mean we can't nerd out on it.

One of the core philosophical problems surrounding Trek's transporter is an issue of consciousness and identity: If the transporter takes all the atoms that make up a person, encodes them, beams them somewhere else, and then reassembles them, how can we know that the resulting "person" is the same person who went in? It's easy to argue that the transporter is effectively killing the first person, then creating an identical copy of that person with identical memories in the new location. This brings us to the Ship of Theseus paradox—if you replace all the components in a ship over time, is it still the "same" ship? (In other words, if you replace all the atoms in a person, is it still the same person?)

So it gets messy. In the video below, CGP Grey walks us through the various philosophical (and fictional-technological) problems of the transporter using fun, peppy animation. Set aside five minutes and be prepared to change your perception of how Star Trek really works:

Check out this Reddit thread for a bit more discussion of this stuff.

Side note: This also happens to be an excellent instance of parallel creation, as Jake from the YouTube channel Vsauce3 was working on an extremely similar concept at the same time (right down to the Ship of Theseus business)...but for an entirely different reason. Here's Jake's video: