The cast, clockwise from the top: Shinji Ikari, Rei Ayanami, Asuka Langley Soryu, Gendo Ikari, Misato Katsuragi, Ritsuko Akagi. In background: Evangelion Unit 01. Not pictured: Sanity.
Neon Genesis Evangelion (in Japanese, "Shin Seiki Evangelion", which translates as "New Century Evangelion") is a 26-episode science fiction/action/drama anime series from Studio Gainax, which aired on Japanese television in 1995-96. The series was followed in 1997 by Death and Rebirth, a Clip Show revision of the series which condensed many of the series' episodes into an hour-and-a-half timespan (while also expanding a few scenes). Death and Rebirth also featured the first half-hour of The End of Evangelion, a full-length movie that brought the story to a much more definitive (but by no means less controversial) conclusion than the television series did.
A disaster known as Second Impact changed the entire world — Antarctica was annihilated causing global flooding, the planet's axis shifted, and Earth's human population was halved practically overnight. Fifteen years later, fourteen-year-old Shinji Ikari is summoned to the fortress city of Tokyo-3 by his estranged father, Gendo, for a single purpose: Shinji is needed as a pilot for aHumongous Mecha called an Evangelion, used to battle physics-defying beings known as Angels, which threaten to destroy what remains of humanity (though exactly how they plan to do so, and for what reason, is not explained for some time). This relatively standard Humongous Mecha premise gradually transforms into a dramatic character study rife with psychological analysis, religious references, genre deconstruction, social commentary, and exploration of themes such as societal alienation, depression, and the repressive pain of human subjectivity. This approach was unprecedented and revolutionary prior to Evangelion; after Evangelion's runaway success, numerous other anime producers created shows with a similar approach (with varying degrees of success).
As a final note before getting into the tropes: Evangelion is considered Hideaki Anno's brainchild, and it's all but defined his career. Many of Evangelion's themes were directly inspired by his personal battle with depression at the time of its creation; it's unsurprising, then, that he's come to own it, even as it's spun off into numerous extra adaptations which either play on the themes of the anime or ignore them to varying degrees (see the following folder).