
A polyomino is an ncelled "animal" composed of squares connected at their edges. A set of polyominoes can be "free", "onesided", or "fixed", the distinction being the conditions under which two polyominoes are considered distinct. Here I'll explain these distinctions, and illustrate them using the 4polyominoes, or tetrominoes.
When polyominoes are mentioned without qualification, it is usually a reference to "free" polyominoes. A free polyomino is considered unchanged by rotation and reflection. OEIS A000105 gives the number of n celled free polyominoes. Note that the first three tetrominoes pictured here have bilateral symmetry, while the last two do not. Note that the axis of bilateral symmetry can be vertical, horizontal, or diagonal. OEIS A030228 gives the number of n celled free polyominoes lacking bilateral symmetry.

A "onesided" polyomino is considered unchanged by rotation, but not by reflection. In other words, two distinct onesided polyominoes can be mirror images of one another. OEIS A000988 gives the number of n celled onesided polyominoes. The set of onesided polyominoes of a given size consists of the free polyominoes plus the mirror images of the polyominoes lacking bilateral symmetry. In other words, the number of free polyominoes plus the number of free polyominoes lacking bilateral symmetry equals the number of onesided polyominoes. That is, A000105(n) + A030228(n) = A000988(n). The distinction between onesided and free polyominoes isn't important for 1, 2, and 3celled polyominoes because all of these small pieces have bilateral symmetry. This distinction becomes important only for 4celled and larger polyominoes.

Rotations and reflections of "fixed" polyominoes are considered distinct. OEIS A001168 gives the number of n celled fixed polyominoes.

"Positionings" of fixed n celled polyominoes within an n by n square is a concept I think I invented. It comes in handy in the calculation of the number of ways to combine n onesided npolyominoes to make an n by n square. There are 6 ways to position the fixed "T" tetromino in a 4 by 4 square. In general, the number of positionings in an n by n square of a p by q fixed npolyomino (the smallest enclosing rectangle has p rows and q columns) is (np+1)(nq+1). OEIS doesn't contain the sequence of positionings, which for n=0, 1, 2, ... starts 1, 1, 4, 22, 113, ...

For each n, I will present a table of information about npolyominoes.
The first column gives pictures of all the free npolyominoes. Some rows will have more several similar polyominoes.
Free gives the number of free polyominoes presented in this row.
1sided is the number of different ways to reflect the polyomino  1 if it has bilateral symmetry, and 2 otherwise.
Rotations is the number of different ways to rotate the polyomino  1, 2, or 4 depending on its radial symmetry.
Fixed is the number of fixed polyominoes, which is Free * Reflections * Rotations.
Size is the number of rows and columns of the smallest rectangle that can contain the polyomino.
Positionings each is the number of different ways to position each fixed npolyomino in an n by n square. Where the size is p by q, positionings each = (np+1)(nq+1).
Positionings is the number of positionings of fixed polyominoes. Positionings = fixed * positionings each.
By convention, there is one 0polyomino.
0polyomino  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  1  1  0 x 0  1  1  
Total  1  1  1  1 
1polyomino  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  1  1  1 x 1  1  1  
Total  1  1  1  1 
2polyomino  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  2  2  1 x 2  2  4  
Total  1  1  2  4 
3polyominoes  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  2  2  1 x 3  3  6  
1  1  4  4  2 x 2  4  16  
Total  2  2  6  22 
Tetromino  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  4  4  2 x 3  6  24  
1  1  2  2  1 x 4  4  8  
1  1  1  1  2 x 2  9  9  
1  2  2  4  2 x 3  6  24  
1  2  4  8  2 x 3  6  48  
Total  5  7  19  113 
Pentominoes  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  1  1  3 x 3  9  9  
1  1  2  2  1 x 5  5  10  
1  1  4  4  2 x 3  12  48  
3  1  4  12  3 x 3  9  108  
1  2  2  4  3 x 3  9  36  
1  2  4  8  2 x 3  12  96  
3  2  4  24  2 x 4  8  192  
1  2  4  8  3 x 3  9  72  
Total  12  18  63  571 
Hexominoes  Free  1 sided 
Rota tions 
Fixed  Size  Position ings each 
Position ings 
1  1  2  2  1 x 6  6  12  
1  1  2  2  2 x 3  20  40  
2  1  4  8  2 x 4  15  120  
1  1  4  4  2 x 5  10  40  
3  1  4  12  3 x 3  16  192  
2  1  4  8  3 x 4  12  96  
1  2  2  4  2 x 4  15  60  
1  2  2  4  2 x 5  10  40  
3  2  2  12  3 x 4  12  144  
3  2  4  24  2 x 4  15  360  
3  2  4  24  2 x 5  10  240  
4  2  4  32  3 x 3  16  512  

10  2  4  80  3 x 4  12  960 
Total  35  60  216  2816 
Tetriswiki: Square_Platforming
OEIS: A000105 — number of polyominoes with n cells  1, 1, 1, 2, 5, 12, 35, ...
OEIS: A000988 — number of onesided polyominoes with n cells  1, 1, 1, 2, 7, 18, 60, ...
OEIS: A001168 — number of fixed polyominoes with n cells  1, 1, 2, 6, 19, 63, 216, ...
Wikipedia: Polyomino is a clear description of polyominoes — free, onesided, and fixed.
Tetris enumerates the ways to tile an nbyn square by onesided npolyominoes
Tetromino Soup is a puzzle asking for the probability of forming each of the five free tetrominoes by a random walk
The webmaster and author of this Math Help site is Graeme McRae.