Tetromino

A tetromino is a polyomino or block made of four square blocks. The seven one-sided tetrominoes are I, O, T, S, Z, J, and L. The spelling "tetromino" is standard among mathematicians. The Tetris Company had preferred tetramino around 1999 and has used Tetrimino since 2001.

The basic tetrominoes
Each tetromino has a letter name and several alternative names. In most games other than Tetris for NES, Tetris & Dr. Mario, or black-and-white systems, each tetrimino has a different color. However, before about 2001, these colors varied from product to product, making it difficult for people who play different games to discuss "the blue piece". Since then, the color schemes associated with SRS and TGM Rotation have dominated new games, and "the blue piece" is always J.

I
Other names include straight, stick, and long. This is the only tetrimino that can clear four lines outside of cascade games.

Note that this tetrimino is not found in Tetris Best 2 or Tetrinsanity.

O
Other names include square, block, sun and ass. (link --ages 30+, contains sex references, nsfw)

T
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Used for T-spins. Also called the middle finger (link --ages 30+, contains sex references, nsfw) or the Tetris logo. This is not in Tetrinsanity.

S
Other names include inverse skew and right snake.

Z
Other names include skew, reverse S and left snake.

J
Other names include gamma, inverse L, reverse L, or left gun.

L
Also called inverse J, reverse J, and right gun.

Multimino
In The Next Tetris, Sticky mode of Tetris Worlds and Tetris Mania, and Jewelry Master, some tetrominoes are multiminoes, that is, the Tetromino itself is divided into visible components. Though the Tetromino is moved and rotated as a unit, once it locks, the components separate by colors and fall.

Trivia

 * Piotr Grochowski has banned instances of the letter name being connected to the number of minoes (most of the time, 4, so tetrimino) with a dash.
 * That's just the beginning. Piotr has banned everything from periods and commas inside quotes (as it would imply that the name literally contains the period), to i-notation for all block sizes except tetrimino.

Tetrimino vs Tetromino

 * Tetrimino is trademarked (although some fan games also use it); tetromino is generic.
 * A tetrimino is colored in a specific way (although some fan games use a different scheme); a tetromino is not colored in that way.
 * There are 7 tetriminoes; there are 5 tetrominoes (7 one-sided tetrominoes).

Tetrimino (uppercase T) vs tetrimino (lowercase T)

 * Tetrimino is the official name, tetrimino (lowercase T) is the one used in some fan games.
 * A Tetrimino is colored in an official scheme; a tetrimino is used in some fan games and has a different scheme.

Tetriminos vs tetriminoes

 * "Tetriminos" is the official plural, "tetriminos" is used by TB, and "tetriminoes" is used by TBII.
 * Both "tetriminos" and "tetriminoes" are unofficial names.