Hand Seals

Hand Seals, aka kuji-in (literally means "nine syllables seal") are hand gestures that are commonly used as a "mantra" (magic formula). This has roots in Taoism, or possibly Buddhism (as the kuji is said to come to China from Tibet). {Technically, mystic Hand Signs are Mudras, not Mantras, which are spoken. Of course, ninjas, and others who use them often also Call their Attacks, so fall into Mantras as well.}

This trope is popular in Japan (and China), as they're the source of the trope. However, you can expect some works from other countries feature this as well, often parodied. One of the Ninja cliches (along with smoke bombs, kabuki stagehand costumes, and shurikens), but sometimes monks also do this as well, especially when meditating.

The mostly used hand gesture in works is the index and middle finger of both hands pointed up, while the ring and little finger wrap the other hand, as shown above. Another variant is the clap palms, more commonly used by monks. Sometimes a one-hand variant is also used.

Subtrope of Magical Gesture. Compare Pstandard Psychic Pstance, Asian Rune Chant.

Examples of Hand Seals include:

Anime and Manga

Commonly used to do jutsus in Naruto, with the hand gestures represent the 12 Eastern Zodiac motifs (the picture above, in particular, represents "sheep/ram"), but other variants also exist, such as Naruto's crossed index and middle fingers when he use the Shadow Clone Jutsu.
Bleach characters sometimes (but not always) do this when using kido spells. Some examples: Hacchi using Bakudo #99, Rukia using Bakudo #9, Shuhei Hisagi using Bakudo #62.
Sometimes used in Fullmetal Alchemist with alchemy.
In Sailor Moon Hino Rei/Sailor Mars uses these combined with a paper charm to perform Akuryo Taisan, which paralyses enemies. She can use it as either Sailor Mars or Rei Hino as it has to to with her being a Miko.
Averted in Ghost Hunt, where the Nine Syllables are used for exorcism, but not with the appellated seals. (A more generic two-fingered gesture is used instead)
Marron and Carrot from Sorcerer Hunters explicitly notes that the Eastern-style magic Marron uses requires two hands to perform the necessary seals in order to cast magic, contrast to the Western-style magic utilized by Sorcerers.

Comic Books

Empowered: Ninjette uses hand seals to stave off a hangover, of all things.
In the G.I. Joe comics Storm Shadow would lead the Joe Ninja team through a series of hand seals as part of an exercise to clear the mind before heading into battle.

Film

In Ninja Assassin, the main character makes use of these in order to close up some gaping wounds. They Did the Research, too, using the full set of kuji-in hand seals in the proper sequence.
In Scott Pilgrim vs. the World, Gideon does this to summon his uber-katana.

Literature

In the The Witcher Witcher series, witchers are capable of using 'signs' by forming a specific shape with their fingers. The gesture itself seems to be very important because in the first game one of the younger witchers explains that he is unable to use signs because he broke his fingers during training.

Live Action TV

In Kousoku Sentai Turboranger, Zimba does this sometimes when enlarging a monster.
- And in the earlier Hikari Sentai Maskman, the heroes use Hand Seals to transform.
And of course it reigns supreme in the Power Rangers and Super Sentai series with a ninja theme or ninja-themed bad guys. Bonus points for a smaller robot using hand seals to make the bigger robots combine, and a Butt Monkey comedic villain stealing from a ninja Big Bad by imitating the ninja gesture to make a door open.
Spoofed in Kamen Rider Fourze at the opening: Fourze turns invisible with the Stealth Switch, but he really didn't need to do a hand seal to do it.

Video Games

Hanzo Hattori and Kotaro Fuuma from Samurai Warriors both use this kind of spell, ususally in order to lock/unlock a gate.
Phantom, the ghost ninja robot from Mega Man Zero uses this to do the Doppelganger Spin.
- Cubit Foxtar of Zero 3 uses a one-hand variant to summon his 9 Attack Drones.
Guy, of Final Fight and Street Fighter fame, is sometimes seen performing one-handed seals (like Haku, but he predates him by about a margin of a decade) when performing moves such as his Bushin Senpuukyaku or Bushin Musou Renge. Two of Guy's Victory Poses in Street Fighter Alpha consist of 'em (one them with his back to the player). In Super Street Fighter IV, two of his personal actions, his Focus Attack, and his end of match win pose also involve hand seals, in addition to the above.
- Ibuki also does this when performing a Smoke Out in her Victory Pose, as well as before a few of her supers.

Web Animation

Dead Fantasy 2. Ayane does this when she jumps to the top of the tower and starts casting a spell. Watch it here, starting at 2:15.

Western Animation

Batman: The Animated Series: Kyodai Ken frequently uses this.
Fridjitzu requires this.