(part of common anime words)
Honestly, it's easy to get confused. The word omake is one of those that get thrown around by fans a lot but nobody ever says what it really means. An omake おまけ is, basically, something which is "extra" or that you get besides something else.
For example, say you buy a burger at some fast food and you get an ice cream for free with your order. That ice cream, it "was an omake," omake datta おまけだった.
But that's not all. An omake can also be something you get extra that is not good at all and you don't really want. Like say your best friend makes a pun so bad you end up killing him for it, you lose your best friend and "on top of it you become a criminal," omake ni hanzaisha ni natta おまけに犯罪者になった.
Omake in Anime and Manga
When we talk about omake in anime and manga specifically we are usually talking about content which is either at the start, middle or end of the episode or chapter and that has absolutely nothing to do with it.
Extra information about the world or characters is a kind of omake. Character profiles with age, blood type, favorite food, hobbies, etc. are omake. Those "comics with four panels," yonkoma 四コマ, are omake. Even illustrations and pin-ups that would be hard to put in the middle of the manga or anime story can be omake.
Examples of Omake
In Shingeki no Kyojin 進撃の巨人 (Attack on Titan), details about the 3D maneuver gear and about the walls of the story can be found at the start of certain chapters in the manga. This information is also shown. In the first and second part of some of the anime episodes. Both forms are omake.
In the anime Gintama 銀魂, though it's difficult to say given the nature of the anime, there are some scenes which appear at the start and end of some episodes that can be considered omake. For example, the ones where Gintoki calls himself Ginpachi-sensei and answers certain questions. These take a couple of minutes and usually have nothing to do with the story of the episode itself.
Even games can have omake. The series Metal Slug, for example, used to include drawings of characters and machines of the game that could be unlocked and seen as the player progressed through the game.
One last thing about omake... are omake really "extras"? Are extras really extras? If you pay for something and get it plus something else, haven't you just paid for it and the extra, too? Where does the thing really ends and the omake really starts? Who knows!