This is series of animated shorts is supposedly based on Amethyst, one of the characters from DC Comics, though I don't think this show's protagonist has the same backstory or anything as the original DC Comics character of the same name (and I'm guessing is supposed to be a completely different character, who just happens to have the same name). It is made by DC Entertainment, a brand by the same company that owns DC Comics. This show sadly never made it past its pilot mini series, which consists of 7 episodes, each between 1m16s and 1m20s in length.
The story involves a girl (who appears to be in her early to mid teens) named Amy, who is sent into the world of a video game she was playing shortly after the prince character (the game's protagonist) she's playing gets defeated in the last level, which somehow causes the game to glitch and ultimately pull her into it. In this world, she is Amethyst, princess of Gemworld (which the name of the game's world), and has a magic sword and shield, which appear to actually be made entirely out of energy (or maybe a supernaturally strong amethyst crystal). When she first enters the game world, she has a mohou shoujo style henshin, which turns her into the princess in the game. Unlike most video game princesses though, she takes the fight to the badguys, rather than ending up getting kidnapped. Just because the game's protagonist was defeated before she was pulled in (thus he's a dead when she is sucked into the game, and she becomes the protagonist), does not prevent her from kicking enemy butt. In fact, she fights far better than the game's original protagonist.
In keeping with the video game world concept, each episode is called a "level" in this miniseries. You can watch all 7 episodes on DC Entertainment's Youtube channel. I've directly linked to them for your convenience. Oh, and there's something interesting in one of the episodes. It's a quote from a character in an actual video game that is said by one of the characters in this show. See if you can find the quote. You'll have to be familiar with some older games to get this one.
I just found some interesting info on Wikipedia regarding the original Gemworld comic book. In the original story, the princess's name is Amy, and she is the princess of the Amethyst kingdom in an alternate realm called Gemworld. In the show, Amy is the name of a girl from earth who has a video game called Amethyst: Princess of Gemworld, where the world and kingdom are both called Gemworld (at least I assume this, as multiple kingdoms are not mentioned and the place is simply called Gemworld) and the name of the princess of Gemworld is Amethyst (instead of her name being Amy, and the kingdom's name being Amethyst), which Amy is transformed into (magical girl style) when she gets sucked into the game. In the original Amethyst: Princess of Gemworld series of DC comics, video games do not play any role in the story at all.