No matter what range of emotion you're feeling towards another character, the best way to express it is to say their name—preferably in either a hoarse whisper or scenery-chomping cry. Even if no one's around to hear you, just thinking about that character is enough of a prompt. It's all but required if you're in the midst of rescuing your True Companion/Love Interest currently facing imminent doom, you're running towards each other in romantic rapture, or when your mortal enemy has simply gone too far. Sometimes those last two get confused.
Additionally, if your show features Loads and Loads of Characters this also serves a narrative purpose: When they appear again after not being seen for three arcs, clearly stating their name once more helps the viewer identify and remember them.
Of course there are times (especially in anime) when they overdo it (see Big Word Shout)...
Characters saying their own name can also fall under this trope, but if they do it incessantly, not for dramatic effect, that's Pokémon Speak.
Many instances of this trope in Japanese works may also fall under First Name Basis. In Japanese culture, it is customary to use family names plus an honorific (usually "san", sometimes "senpai" for one's elders, "kun" for underlings, "dono" or "sama" for works set in older times, etc.) when addressing someone else, even directly (they rarely use the pronoun for "you" in formal settings, although they do have a few for informal situations). Children, especially girls, will often use first names with friends, although adults will usually only do this if they've been friends since childhood or are otherwise close buddies. But when a couple has moved ahead in their relationship, they will often switch to using first names as well. In media, the first time a character calls their romantic interest by their first name (whether out of affection or distress), the scene often ends up being an example of this trope.
Can overlap with Angry Fist-Shake.
Compare First Name Basis, First Name Ultimatum and "You!" Squared, Skyward Scream. If you have a lot of people, all doing a "Say My Name" to each other, it's a Rocky Roll Call.
Contrast I Know Your True Name where speaking it has much more dangerous implications.
Examples of this trope are often subjected to Memetic Mutation.
Not to be confused with Spell My Name with a "The".
Earth X: Uatu the Watcher has spent thirteen issues referring to Machine Man, aka Aaron Stack, by his model number, X-51, to emphasize that he is a robot and should have no loyalty to humanity — indeed, no human emotion at all. In the final issue, after being cut off from his sensory feed and left helpless, Uatu calls after him pleadingly: "X-51? X-51, please. ...Aaron?"
Superman shouting Mongul's name after freeing himself of the Black Mercy in "For the Man Who Has Everything". The force of the shout knocks Robin clean off his feet, and lets Mongul know he's only got a fraction of a second before Superman plows into him.
Green Lantern Hal Jordan returns to Earth to find Coast City destroyed by Mongul and the Cyborg. He makes a beeline for where Mongul is beating up Steel and, just like Superman, screams Mongul's name as he tackles him.
In one story the title character's alter ego Bruce Banner makes this comment to a screaming Doctor Doom right after helping to take him down.
Doom had been fighting a Hulk robot that was screaming "DOOOM" as it pummeled him. As Doom turned the tables on the robot, he said "Once more...with feeling...say my name!" There is later an Ironic Echo of this scene.
Shazam: Invoked in Captain Marvel's origin story, when the ancient wizard commands Billy Batson to speak his name: SHAZAM
In the Batman storyline Knightfall, Bane attempts to make Batman scream his name for mercy. Batman tells him "Go back to Hell" and Bane promptly breaks him. Years later, in the one-shot Batman: Bane, Batman finally does shout Bane's name. It's not for mercy, though, it's in frustration - the villain escaped capture.
Used by Empowered during sex. "What's... my... name?"
Inuyasha and Kagome say each other's names many times in InuYasha. This tendency gets lampshaded twice so far in the Continuation FicInuyasha: Beyond Tomorrow; in chapter 15, when Ren grabs Kagome and begins to make off with her, Miroku and Sango get so annoyed at it that they tell Inuyasha to just shut up and save Kagome already, and in chapter 22, when Inuyasha begins to call for Kagome after she jumps down the Bone Eater's Well, Sesshomaru promptly Dope Slaps him and tells him not to start.
Twilight furiously screams Navarone's name in Diaries Of A Madman, after finding out she's been beaten to the Crystal Heart.
During Zuko and Katara's big reunion scene in How I Became Yours, they yell each other's names as they run into each other's arms in the middle of the ballroom.
When England cries out for Scotland in Pottertalia fic Snakeskins, many readers swore they could even hear his voice break with just text.
Probably at least half of the dialogue of WALL•E and EVE consists of their own and each other's names. The third most common word is "directive".
Try keeping track of how often the hero's name is said in Beowulf, most notably in the sea monster fight ("BEEOOWUULLF!"). There are quite a few shouting of "I am Beowulf!" in there (of varying volume and pacing), and Wiglaf even gets the odd "You are Beowulf!".
Disney's Beauty and the Beast has a climactic scene towards the end where Belle calls out "Beast!" The audio commentary notes that they'd forgotten to give the prince a name.
All Dogs Go to Heaven: CHAAARLIE!!! Though it is shouted a few times, the sheer number of times every single character feels the need to say his name is baffling. It averages at one "Charlie" a minute. Have a look.
The subpar Disney movie The Wild had "RYAN! RYAN! RYYYAAAN!" over, and over, and over.
The Shadow does this with The Shadow/Lamont Cranston and Margo:
Former WCW and WWE wrestler Bill Goldberg caused fans to chant "GOOOLD-BERG! GOOOLD-BERG!" during his entrance; combined with his epic entrance music (and his streak), it helped sell him as a Badass.
Current WWE superstar Alberto Del Rio might be the Trope Codifier for Professional Wrestling - at least in the modern era. He has a very hammy ring announcer, Ricardo Rodriguez, that routinely ends every introduction (the vast majority in Spanish, of course), with an absurdly long yell of the wrestler's name. "ALBERTOOOOOOO..." frequently cracks a full ten seconds - and now that he's turned Face within the last year, the vast majority of the crowd participates in the trope as well.
The Hero Shirou and Big Bad Kotomine seem to enjoy saying each others' full name, especially when yelling. More the former than the latter.
Kitomine seems to be fond of saying Shirou's full name, for dramatic effect. All other cases of "Say My Name" in this game are appropriate for the situations but Kitomine loves going out of the way to say "Emiya Shirou" even when a simple "you" would do.
Shirou will also yell "SAABEERR!" a lot, particularly when he summons her during his fight with Rider.
Belkar, who addresses the elf wizard Vaarsuvius as "Elf" or "Ears" or, more often, with some insulting nickname, when he and Haley are left behind after the other half of the team sails away: "OK, elf, enough with the jokes. You got me, fair enough, now bring the ship back. (...) No, seriously, wherever you're hiding: You really had me thinking for a moment that we got left behind. Great illusion. Now, can we just get out of this stupid city already? (...) Vaarsuvius?"
Rose yells STRIIIDEEER! at Dave as her house is about to be meteor'd and she wants him to hurry up. Also, he just threw her bed out of her house and put a lot of important devices in cumbersome, hard-to-reach places.
In Panthera, this is played for drama when Jason/Leo starts doubting everything he thought he knew about himself both in relation to the rest of the team and by himself as a result of his Mentor Mole revealing his true colors and getting killed and failing to come up with a plan in North Korea.
Leo: Cats don't need licenses to hunt, fight, kill. But they never have to worry about why they do it, or the greater consequences, either. And we... Tigris, what's my name?
Tigris: I... don't follow.
Leo: My name. What's my name?
Tigris: Uh. "Leo"?
Leo: No. "Leo" is the lion mask I wear into battle. And yes, the mask is deeply a part of me. But my mind — my soul — is human. My name is Jason, and it's me who has to live with the things I do.
In Nullmetal Alchemist the moment Ed hears Majhal's name he begins screaming it, freaking out everybody around him.
In the Bad Days short about The Walking Dead, most of Rick's dialogue consists of calling out, "Lori! Carl!" Lee's only line involves yelling, "Clementine!" Finally, after Rick discovers that Lori has cheated on him with Shane, Rick growls, "Shannnne!", then beats Shane up.
In The Autobiography of Jane Eyre, vlog version of Mr Rochester likes saying Jane's name a lot. It's very clear especially in episode "Consequences". She calls him "sir".
The alien brain Krang in Teenage Mutant Ninja Turtles (1987) liked watching a TV show that seemed to consist entirely of a man named John and a woman named Marsha staring at each other and saying their names over and over.
MAD: John didn't have his glasses on, and was saying it to the wrong girl at first. Also, he stepped off a cliff to kill himself; a trampoline saved him (repeatedly); the John-Marsha scene played; and when Marsha rejected him, that was the point when the trampoline people moved the trampoline...
There exists a piece of fanart of L and Light from Death Note having sex. Light says the trope, and it takes L a few repetitions to realize he's screaming "Kira".
During Muhammad Ali's fight with Sonny Liston, who would only refer to him as Cassius Clay (his old name), Ali repeatedly shouted, "What's my name? What's my name?" A subsequent opponent of Ali's did the exact same thing. So did Ali, dishing out the most one-sided beating of his career.