In the Iron Man animated series, why does teenage Tony Stark have an arc reactor, but then in Iron Man he doesn't get one until a little bit in?
tl;dr: Because the two Tony Starks are different characters.
The Marvel comics universe is huge; so big that it can't even fit into a single universe. There have been so many different versions, revisions, reboots, and adaptations of those characters, that it would be impossible to come up with a single backstory for anyone, let alone someone as common as Tony Stark.
Instead, Marvel's stories are split up into different "universes" within the larger "multiverse". Each universe within the multiverse exists independent of the others, with its own history, characters, sometimes even its own laws of physics. This concept has been explored several times in the comics, most recently with the Secret Wars storyline that kicked off this year, showing all the various universes colliding with each other.
Iron Man, the movie, takes place in the Marvel Cinematic Universe, which includes all of the live-action movies and television shows produced by Marvel/ABC/Disney (that includes all of the Avengers-related movies, plus Guardians of the Galaxy, as well as Agents of SHIELD, Agent Carter, and the Netflix series like Daredevil). Officially, this is Earth-199999. Again, all of those movies and TV shows are internally consistent, but they have nothing to do with the animated series.
The various animated series, on the other hand, generally exist within their own unique, or very limited, universes. Some examples of Tony Stark in various animated series:
- Iron Man: Armored Adventures takes place in it's own unique universe, officially called Earth-904913. This is the only show set in that universe, which is how Tony Stark can be Iron Man as a teenager.
- Iron Man: The Animated Series from 1996 exists within the universe for the Marvel Action Hour (which also includes 1996 The Incredible Hulk animated series, designated Earth-534834.
- A group of Marvel animated shows starting in 2010, including Marvel's Avengers Assemble among others, take place in a universe unofficially referred to as Earth-TRN123 (because it has no official designation yet.)
On a related note, the majority of the comic book issues take place in Earth-616, the "mainstream" universe, while the Ultimate whatever issues take place in Earth-1610, the "Ultimate" universe. That's how, for example, Tony Stark could invent Ultron in the movies, but Hank Pym invented him in the comic books.