A character who is In Training for something, usually under some kind of Mentor. Since they're still just learning, they are usually not very good, but if the show is long enough, they'll improve at least a little. Unless they're a Butt Monkey. If they're in a movie, they'll improve after a Training Montage. Sometimes have a chance of falling and becoming a villain, sometimes The Dragon or even the Big Bad, and that usually causes Redemption Equals Death.
And, of course, the life goal of some apprentices is To Be a Master.
For the TV Show, go here.
Kenichi. Take a guess on how much a teenager can learn about martial arts in 39 volumes and counting.
Yahiko in Rurouni Kenshin. He does progress, although rather slowly over the course of the many volumes of the manga and accompanying TV series. At the beginning of the series, Yahiko is most useful to the heroes as a thrown weapon (it happens). At the end of the series he has become a master swordsman in his own right.
Elena in The Fairy Godmother by Mercedes Lackey ends up apprenticed to, well, a fairy godmother. Comparatively little of the book is actually devoted to her apprenticeship, however; the plot doesn't really kick in until the job has officially been handed over to her.
In World Without End, Merthin, as an apprentice carpenter, is actually much smarter and more competent than his master and Arch-Enemy Elfric.
Jamie ends up as an apprentice in the Clown Division of The Pilo Family Circus. Because of the mishaps that beset the clowns, he doesn't get to perform much, but he still ends up better off than the previous apprentice...
The protagonist of the Rivers Of London series of books, Constable Peter Grant, is a fully qualified policeman as of the first book, but also an apprentice wizard indentured to Detective Chief Inspector Nightingale. There used to be a Wizarding School but the birthrate of people who could use magic dropped making it unsustainable that and most of the people who would've been able to teach magic were killed during World War II.
Shintaro Gotou in Kamen Rider OOO becomes one to Akira Date/Kamen Rider Birth, eventually using a second Birth Buster to back Date up. Differs from the Movie War Core continuity, where Date does not appear and Gotou becomes Birth.
In early editions of Dungeons & Dragons it was customary for mages/wizards to learn by becoming apprentices of established mages/wizards. A number of wizard/apprentice relationships appeared in the Forgotten Realms setting. For example, Elminster has had many apprentices over his long life span (hundreds of years).
In Warhammer 40000 Interrogators are apprentices to Inquisitors. They are slightly more experienced than is usual for the trope; by the time you are even considered for the rank of Interrogator you have probably been serving as a Throne Agent for several decades.
In The Incredibles, Buddy is a perfect example of the Apprentice becoming the Big Bad. He idolizes Mr. Incredible and attempts to force his way into a role as his sidekick. When he is spurned by the hero, he becomes bitter and attempts to play a hero by causing catastrophes that only he can fix.
Chowder is one of these to Mung Daal; other child characters are apprenticed to other chefs.
Brainy Smurf is this to Papa Smurf on The Smurfs; likely not the best decision Papa Smurf ever made, and at least on one occasion, he scolds Brainy and says he "has the right mind" to get a new apprentice after Brainy causes a disaster.
Although Terry was technically Batman in Batman Beyond, he could still be considered this to Bruce Wayne. (Lampshaded in one episode where Bruce was getting treatment from the Lazarus Pit and the possiblity of him becoming Batman again was brought up; Terry said firmly that he was "not wearing the Robin outfit".)
Before the age of streamlined education most (if not all) professions were learned by apprenticing under a master. Children usually apprenticed under their parent until said parent was too old to work, at which point the child would take over the business. This is the origin behind such surnames as Smith, Carpenter, and many other last names that sound like jobs.