You have Mirror Monologue, where the character is talking to his reflection.
And then you have this trope, which is when the reflection is the one doing the talking (or, occasionally, moving).
The difference? The Mirror Monologue is realistic (physically, if not psychologically); nothing fantastic is happening, so we can pretend we're objective observers. Many versions of this explicitly have the mirror moving in a completely different way than the 'real world', thus bringing it over into either the fantastic, or we're explicitly seeing things through the character's eyes. Those that don't do the 'mirror isn't reflecting the real world' thing via special effects of one sort or another but still qualify are using editing for much the same effect (see, e.g., the first Spider-Man movie, or the Gollum/Sméagol dialogue from film version of The Two Towers).
Usually used, much like the Mirror Monologue, to make visual an internal dialogue, either between different aspects of a character (such as a man and his conscience) or between different personalities sharing the same body.
Somewhat of an inversion of Mirror Routine; there, it's a different person acting like a reflection; here, it's a reflection acting like a different person.
It happens more than once to Allelujah in Mobile Suit Gundam 00. For example, in the last episode of the first season, he has a conversation with his other personality, Hallelujah, that is reflected in his Mobile Suit's screens.
One episode of Betterman has this happen to the entire main cast besides Keita and Hinoki. Each reflection delivers a "The Reason You Suck" Speech, but is eventually defeated by a Shut Up, Hannibal! counter-argument. For the record, these were hallucinations somehow tied to Algernon.
Rockula has recurring scenes where Ralph converses with his reflection, who moves independently of him and is implied to even have a life of his own. It's never really made clear if this is all in the character's head or if this is actually happening - He is a vampire in a film where Our Vampires Are Different.
Once when Mick Jagger was the musical guest he did a sketch where he talked to his mirror self, played by Jimmy Fallon.
This got a Continuity Nod and a twist when Jimmy Fallon guest-hosted in 2011 and did the same basic sketch with Andy Samberg playing his mirror self.
Averted when Prince was musical guest during the time that Fred Armisten had as a recurring sketch "The Prince Show" where he played Prince having a talk show. They did a mirror bit but he just looked at himself. Could be a Subversion (in that everyone expected a mirror bit so they did that), or that they wrote a mirror bit and Prince nixed the idea, so they had to replace it.
A skit on The Muppet Show had Gonzo singing "Act Naturally" when his reflection suddenly starts singing along and then arguing with him. (You know it's unusual when even Gonzo is shocked by it.)
Similarly on Muppets Tonight Gonzo performed "Dancing With Myself" in a mirror room. In this case the reflections exited the mirrors.
In the beginning of the Dinosaurs episode titled "How to Pick Up Girls" Robbie has a borderline argument with himself in the mirror on the inside of his locker door.
Niki and her alter-ego, Jessica, would often interact this way on Heroes, with the dormant personality on the other side of the mirror.
Criminal Minds had an Unsub of the Week, who was being egged on to kill by hallucinatory people, see his reflection move independently and join his other hallucinations in urging him to run when the police were coming.
All My Children. Janet Green's reflection would talk back to her whenever she was going of the deep end.
Subverted in Pogo. It looks like Porkypine's reflection is talking back to him, but it's actually a bug hiding behind his mirror playing a prank on him.
A comics gag seen in both Doonesbury and Bloom County. (B. Breathed later said he inadvertently ripped off the joke from Trudeau).
Some early 60s The Perishers strips had a Running Gag where the neurotic Wellington would pour out his existential anguish in front of his mirror, and his reflection would make some sarcastic remark once he'd left.
Used in FoxTrot, of all places, when Jason complains to his reflection how unfair it is that his family want him to donate his allowance to hurricane relief rather than buying a newspaper comic collection. His reflection agrees and goes on about how unfair it is of the family to assume that the needs of those whose homes and lives have been destroyed outweigh his desire to giggle for 30 minutes (or 45 if he reads slowly). Jason's final comment is "I'm not sure I like what I'm seeing".
In Sly 3: Honor Among Thieves, once enemy-now-ally Panda King has a discussion with his former self from the first game. Panda King needs Sly's help saving his daughter, but is still angry over his previous defeat. The two halves, anger and humility, reconcile when the game-3 Panda King points out that the two can work together to save Jing King, and become the father she needs. ("The yin..and the yang?")
In another Disney short, "The Golden Touch", King Midas brags to his reflection that he will turn everything to gold, and the reflection applauds him. Later, as he realizes he's Blessed with Suck, he asks his refection if he's doomed to starve to death. The reflection turns into a golden Grim Reaper and nods yes.
In an episode of Foster's Home for Imaginary Friends, a feverish Bloo halucinates that his reflection is talking back to him. When his friends mistake Bloo for a ghost, his reflection even yells to run for it.
In an episode of The Mask animated series, the Mask does this after being framed for stealing cookies from orphans and doubting his own innocence. Of course since he's a cartoonish Reality Warper:
The Mask: But...I didn't do it. (stares at reflection in a nearby window) Reflection:(leaning out of the glass) How do you know? Maybe you have one of those split personalities.
Appears in a few episodes of Sponge Bob Square Pants. In "Wet Painters", SpongeBob is telling himself to be brave, but the reflection says "You're on your own" and leaves. In "Something Smells", the reflection breaks the mirror to get away from SpongeBob's bad breath, leading him to believe that he's Mirror-Cracking Ugly.
"Roller Cowards" has a scene that involves Patrick literally punching out his own reflection after it gets into a short argument with him.
Patrick: I thought we settled this the last time.
A creepy version occurs in "Rule of Dumb" where Patrick speaks to a monstrous version of himself in the mirror.
In The Super Mario Bros Super Show episode Mario and Joliet, A Romeo and Juliet parody, King Koopa learns that Princess Toadstool has halted the feud (Which he started!) between the fighting families, and as a result, he can't sell them weapons anymore. His reflection tells him to just kidnap the Juliet and blame it on the Romeo, setting the families against each other again.
Happens in Jimmy Two-Shoes, when Jimmy is feeling guilty about running off and not standing up for Beezy.
In the Musical Episode of Daria, Quinn sings in front of a mirror while trying to decide what to wear. Her reflection tells her she could pick any outfit and still look attractive.
In The Twisted Tales of Felix the Cat ...after Felix's reflection reaches out of the mirror to help spruce him up for a date, no less. (Not surprising, since the show runs on 100% old-school cartoon logic.)
Felix's Reflection: Felix, ya look like a million bucks!
Felix: Thanks, Felix. Wish me luck, man! (leaves)
Felix's Reflection:(sighs) Poor mook don't stand a chance.
Parodied in The Simpsons episode "My Fair Laddy", in which Willie sings and his reflection joins in. Willie freaks out and smashes the mirror.
Pepper Ann did this so often, some viewers could consider Pepper Ann's reflection a side character in her own right.
Teen Titans Cyborg loses to Atlas who imprisons his friends as his trophy's, his reflection in the glass talks to him into fighting Atlas again, the conversation switches from Cyborgs human side, and his mechanical side.
Ultimate Spider-Man after an exhausting day of fighting the Frightful Four, Peter totally forgot to buy a cake for Uncle Ben's birthday(whose dead). His Spider-man reflection talks him into taking Fury's offer to train him to be the Ultimate Spider-Man.
In the Ninjago episode "Wrong Place, Wrong Time", the ninja go back in time to the pre-series special and accidentally alter the timeline by interfering with Kai, Nya, and Wu's past selves fighting Lord Garmadon's army. The result is that the skeletons don't capture Nya, which is pivotal to Kai's drive to become a ninja, so the ninja devise a plan to hand Nya to the skeletons which involves Kai having to pretend to be his past self's reflection to distract him while the other ninja grab Nya.
A variation occurs in Minnie the Moocher, where Bimbo looks into a well and his rippling reflections do a "Hi-de-ho" in tune to the song.
A rather ominous example takes place during the Will Vinton claymation adaption of Rip Van Winkle. After Rip washes his face when arriving in the mountains to go squirrel hunting, he laughs at his reflection in the water and asks, "And what are you doing here?" It then asks the same question in an echoey fashion, prompting Rip to run away fast.
The new Mickey Mouse short "Eau De Minnie" has Minnie's reflection warning her not to overdo it on spraying perfume, but Minnie flips the mirror to silence her.