You may have seen this facial gesture in Japanese anime, manga, or in the movies. While it is considered immature and childish, it has a rude or dismissive meaning–not as bad as the Western middle finger, but not polite either. It’s a way to mildly insult someone. It is called akkanbe- and involves facing someone while using a finger to pull on the lower eyelid and (usually) stick out the tongue.
The earliest reference to this act is in a novel by Katai Tayama. He says the origin is a corruption of 赤い akai (red) and 目 me (eye). Makes sense since pulling the lower eyelid reveals a red area.
Albert Einstein sticks his tongue out in an akkanbe- fashion to the right. One comic of the robot cat, Doraemon, has アカンベーダー akanbe-da- which is a parody of Darth Vader. Crayon Shinchan also employs akanbe at times.
This can be written/pronounced as あかんべえ (akanbee), あっかんべー (akkanbe-), or, in katakana, アッカンベー (akkanbe-).
If you want to show your love of Japanese while being a bit rude about it, we have a few akkanbe- themed products over at TheJapanShop.com. Check them out.