The different rule descriptions use common phrases which are explained in detail on this page.
Neighborship: A cell (red) has 3, 5 or 8 neighbors (blue and green). The blue cells are the orthogonal (horizontal and vertical) neighbors of the red cells. The green cells are the diagonal neighbors of the red cells 

Regions and areas: A region consists of one or more cells which are separated by bold lines from other regions. An area consists of one or more cells which are not separated by bold lines from other areas, but defined otherwise, e.g. all adjacent cells of the same color. In the diagram on the left the yellow cells form such an area. Regions as well as areas can be square, rectangular or irregular. (Note: A square is also a rectangle!) 

Orthogonally contiguous cells/areas The white cells form an orthogonally contiguous area, if it is possible to travel from any white cell to any other white cell, moving only horizontally and vertically (never diagonally) and never crossing a black cell. In the diagram on the left the white cells are colored blue, green and yellow (actually, they all are white). • All green cells are orthogonally contiguous: For example, one can travel from A to B moving only horizontally and vertically, not crossing a black cell. • Also, all blue cells are orthogonally contiguous. • The green cells and the blue cells are not orthogonally contiguous. For example, there is no path from C to D, travelling only horizontally and vertically which does not cross a black cell. • Cell E is isolated and is not orthogonally adjacent to any other white cell. Orthogonally contiguous areas of black cells are defined analogously. 

Manhattan Distance (Taxi Distance) The Manhattan Distance is the shortest link between two cells, travelling only horizontally and vertically between adjacent cells. The image on the left shows three possible paths; all paths are equally in length. Formally: The Manhattan Distance between two cells is the sum of the absolute differences of the coordinates of the cells the path travels through: In the example on the left the Manhattan Distance between the two blue cells is 7 (= number of cells the path goes through including the target cell). 

Dominos A domino is a region of exactly two orthogonally adjacent cells. Disregarding rotation and reflection there is exactly one domino: In the diagram on the left you can find 3 Dominos. Yellow and blue have the same orientation; yellow and green have different orientation. Yellow and green are orthogonally adjacent; yellow and blue are diagonally adjacent. 

Triominoes A triomino is a region of exactly three orthogonally adjacent cells. Disregarding rotation and reflection there are exactly two triominoes, labeled by letters according to their shapes: In the diagram on the left you can find 3 triominoes. The orange and the blue triomino together cover a 2x2 area which is not allowed in many puzzle types. 

Tetrominoes A tetromino is a region of exactly four orthogonally adjacent cells. Disregarding rotation and reflection there are exactly five tetrominoes, labeled by letters according to their shapes: In the diagram on the left you can find 3 Tetrominos. 

Pentominos A pentomino is a region of exactly five orthogonally adjacent cells. Disregarding rotation and reflection there are exactly twelve pentominoes, labeled by letters according to their shapes: 

Polyominoes (Nominoes) A polyomino is a region of orthogonally adjacent cells; an Nomino is a region of exactly N orthogonally adjacent cells. Dominoes, triominoes, tetrominoes and pentominoes are special cases of Nominoes (N=2, 3, 4 or 5). 