If you watch much anime, you know what it means to 'level up'. I imagine the term comes from the gaming world originally, where it means you clear a level and get to the next one. But "level up" is used far outside the gaming world now. Anyone who in any way clears an obstacle and moves forward can be said to have leveled up.
A LOT of anime series contain plot lines which require their characters to achieve new levels of battle abilities. Some of these plot lines involve continuous leveling up and continuous growth of the character's abilities thoughout the series, and some of them only have the main characters level up one major time, or perhaps a couple of times
Bleach is a series that has its main character, Kurosaki Ichigo, undergo the most continuous level ups of any series I've watched. Ichigo achieves his first level up when he takes Rukia's powers in the first episode and begins to use shinigami powers. From there the process is continuous. His major level ups include: regaining his powers after his soul chain is cut, attaining bankai, mastering his hollow side and fighting masked, achieving the state necessary to bring Aizen down, and then regaining his powers using Fullbring. He also has a multitude of lesser level ups along the way, including each one he achieves as he fights Renji, Kenpachi, Byakuya, GrimJoe and Ulquiorra. The entire series follows Ichigo as he struggles to attain new levels of fighting skills in order to protect the people he cares about.
Inuyasha is another series that follows the relatively continuous growth of a single character's battle abilities. Inuyasha, the series main character, is a half human, half youkai who inherits a sword early in the series. Throughout the series the sword's abilities and his ability to use the sword continuously level up. He learns to use the sword's true ability along the way, gains the ultimate technique with it, gains an ability of the sword to break barriers when in a red-bladed form and eventually gains two more higher level abilities with the sword. Like Ichigo in Bleach, at each level he is faced with characters or circumstances beyond his current abilities and must level up to 'clear' the difficulty and move forward.
Some series have essentially a single major level up event, and examples of this type would include Rurouni Kenshin, D.Gray-Man and Nurarihyon no Mago. In Rurouni Kenshin, Kenshin must work to attain the ultimate technique of his Hiten Mitsurugi sword style before being able to face his main enemy, Shishio Makoto. Achieving the technique is the major level up he does in the series.
Allen Walker in D.Gray-Man is left for dead by his enemies after losing his Innocence, and he must strive to regain his ability to use Innocence before he can resume his battles against the forces of evil. And no, that's not what it sounds like. If you haven't watched D.Gray-man, Innocence is a mystical power which occurs in a different form in different people and can be used by them to battle the evil Millennium Earl and his minions. Allen spends multiple episodes in this struggle before coming to terms with his Innocence and regaining its use. This is his major level up of the series.
In Nurarihyon no Mago, Nura Rikuo is a half human, half youkai middle school student who find himself as heir to his grandfather's and father's Night Parade of 100 Demons. Although initially he only wants to live as a human with his human friends, when his clan is attacked by other demon clans he eventually must do a major level up that allows him to learn his clan's secret technique and use it to win the battles against the other clans. Although Rikuo has a few smaller level ups along the way, the ability to use the clans secret technique is the one that saves him and his clan and his friends.
Level ups are also used in sport-based or game-based anime. A good example is Hikaru no Go. As Hikaru learns to play the game of Go and plays ever stronger opponents, he has to increase his own skills to move forward. Obviously this type of leveling up isn't quite as intense as the gaining of new skills just ahead of being destroyed that is seen in series like Bleach or D.Gray-man, but pretty much in all anime where this plot device is used, I still find myself thinking, come on, come on, level up!