In the context of anime/manga or JRPG's, it's like "Bonus Material". TV Tropes has a page for "Omake" that's labelled Bonus Material:
A bonus that is added to, but separate from, a particular work. The practice of adding such a bonus to one's work has been around for a long, long time: Charles Dickens, for example, was a regular practitioner of it, back when he wrote his stories for newspapers and magazines.
If this extra material is another story in the universe, it generally doesn't interact with the primary plot of the work, and is seen as "stand-alone" material. Some times that extra material isn't even considered canon, though such additions can add depth and insight to the primary characters if the writer makes an effort to connect the "extra" to the main material.
Other kinds of extras are making-ofs, outtakes, deleted scenes, concept art, feelies and others.
Wikipedia also has an entry for Omake:
Omake often include comedy sketches where the characters behave out of character, break the fourth wall, or subtly address opinions of the fandom known to the writers. Sometimes scenes from the TV show or OVA are humorously re-dubbed. One example, included on the Video Girl Ai DVD, replays scenes from the OVA series with new voice-acting in a rural accent. Other times, the same actors voice a new script that is more sexually suggestive, often ludicrously so. Omake can also consist of non-canonical, and often comedic crossover clips that sometimes occur at the end of episodes of two shows airing concurrently from the same studio, such as recent Kamen Rider and Super Sentai programs.
A screenshot of an anime omake, taken from Gunbuster. Here, chibi versions of the main characters attempt to explain the concept of "Ice II".
For anime, these are often presented in super deformed style, in the same way manga omake often is. For example, Gunbuster features super deformed characters trying to explain what the writers know to be mostly pseudo-science, or talking about their relationships with each other in a way they do not in the series itself, or in the popular anime series Reborn!, after a couple of episodes one of the characters named Haru Miura has an interview called "Haru-Haru's dangerous interview corner" with each of the characters of the anime in chibi forms, the characters' answers to the questions are often something they would never say in the anime or the manga. For live action programs, although not animated, the expressions and sound effects used for comedic purposes can often be inspired by the omake found in the animated mediums.
The term Omake has use also in video games; the Sega game Shenmue II for the Dreamcast had a hidden folder on the game disc labelled "Omake", found by placing the disc into a computer, containing exclusive wallpapers and conception art.
Another example of an omake in popular culture is related to Square's Final Fantasy IX. The secret "Blackjack" minigame after completion of the game is accessed by means of a button combination. The Final Fantasy "Playonline" site has a secrets section for Final Fantasy IX, which requires passwords given in the official Piggyback guide to enter. The password needed to reveal the button combination for the Blackjack minigame is E-OMAKE. The minigame itself is, of course, an omake.