A stereogenic center (or stereocenter) is an atom such that swapping the positions of two ligands (connected groups) on that atom results in a molecule that is stereoisomeric to the original.  For example, a common case is a tetrahedral carbon bonded to four distinct groups a, b, c, and d (Cabcd), where swapping any two groups (e.g., Cbacd) leads to a stereoisomer of the original, so the central C is a stereocenter.  Many chiral molecules have point chirality, namely a single chiral stereogenic center that coincides with an atom. This stereogenic center usually has four or more bonds to different groups, and may be carbon (as in many biological molecules), phosphorus (as in many organophosphates), silicon, or a metal (as in many chiral coordination compounds). However, a stereogenic center can also be a trivalent atom whose bonds are not in the same plane, such as phosphorus in P-chiral phosphines (PRR′R″) and sulfur in S-chiral sulfoxides (OSRR′), because a lone-pair of electrons is present instead of a fourth bond.
Similarly, a stereogenic axis (or plane) is defined as an axis (or plane) in the molecule such that the swapping of any two ligands attached to the axis (or plane) gives rise to a stereoisomer.  For instance, the C-symmetric species 1,1′-bi-2-naphthol (BINOL) and 1,3-dichloroallene have stereogenic axes and exhibit axial chirality, while (E)-cyclooctene and many ferrocene derivatives bearing two or more substituents have stereogenic planes and exhibit planar chirality.
Chirality can also arise from isotopic differences between atoms, such as in the deuterated benzyl alcohol PhCHDOH; which is chiral and optically active ([α] = 0.715°), even though the non-deuterated compound PhCHOH is not.
If two enantiomers easily interconvert, the pure enantiomers may be practically impossible to separate, and only the racemic mixture is observable. This is the case, for example, of most amines with three different substituents (NRR′R″), because of the low energy barrier for nitrogen inversion.
It is not necessary for the chiral substance to have a stereogenic element. Examples include certain helicenes, calixarenes and fullerenes, which have inherent chirality.  Moreover, it is possible for a molecule to have a center of chirality that sits in a position that does not correspond to an atom. Such a molecule may be chiral without having any stereogenic centers. Examples include 1,3,5(,7)-substituted adamantanes such as (1S,3R,5R,7S)-3-methyl-5-phenyladamantane-1-carboxylic acid (pictured).
When the optical rotation for an enantiomer is too low for practical measurement, the species is said to exhibit cryptochirality.
Chirality is an intrinsic part of the identity of a molecule, so the systematic name includes details of the absolute configuration (R/S, , or other designations).