Intuitive definition: real group of rotations + reflections.
Mathematical definition: group of orthogonal matrices.
Has 2 disconnected: pieces: one with determinant +1 (which is is a subgroup known as the special orthogonal group) and the other with determinant -1.
If a reflection is done, the determinant is -1.
Note however that having the determinant plus or minus 1 is not a definition however: there are non-orthogonal groups with determinant plus or minus 1. This is just a property.
A low dimensional example: