A double bond between two carbon atoms forces the remaining four bonds (if they are single) to lie on the same plane, perpendicular to the plane of the bond as defined by its π orbital. If the two bonds on each carbon connect to different atoms, two distinct conformations are possible, that differ from each other by a twist of 180 degrees of one of the carbons about the double bond.
The classical example is dichloroethene , specifically the structural isomer  that has one chlorine bonded to each carbon. It has two conformational isomers, with the two chlorines on the same side or on opposite sides of the double bonds plane. They are traditionally called cis (from Latin meaning "on this side of") and trans ("on the other side of"), respectively; or Z and E' in the IUPAC recommended nomenclature.  Conversion between these two forms usually requires temporarily breaking bonds (or turning the double bond into a single bond), so the two are considered different configurations of the molecule.
More generally, cis–trans isomerism (formerly called "geometric isomerism") occurs in molecules where the relative orientation of two distinguishable functional groups is restricted by a somewhat rigid framework of other atoms.
For example, in the cyclic alcohol inositol  (a six-fold alcohol of cyclohexane), the six-carbon cyclic backbone largely prevents the hydroxyl  and the hydrogen  on each carbon from switching places.  Therefore, one has different configurational isomers depending on whether each hydroxyl is on "this side" or "the other side" of the rings mean plane.  Discounting isomers that are equivalent under rotations, there are nine isomers that differ by this criterion, and behave as different stable substances (two of them being enantiomers of each other). The most common one in nature (myo-inositol) has the hydroxyls on carbons 1, 2, 3 and 5 on the same side of that plane, and can therefore be called cis-1,2,3,5-trans-4,6-cyclohexanehexol. And each of these cis-trans isomers can possibly have stable "chair" or "boat" conformations (although the barriers between these are significantly lower than those between different cis-trans' isomers).
Cis and trans isomers also occur in inorganic coordination compounds, such as square planar  complexes and octahedral  complexes.
For more complex organic molecules, the cis and trans labels are ambiguous. The IUPAC recommends a more precise labeling scheme, based on the CIP priorities for the bonds at each carbon atom.