Stereoisomers have the same atoms or isotopes connected by bonds of the same type, but differ in their shapes – the relative positions of those atoms in space – apart from rotations and translations.
In theory, one can imagine any arrangement in space of the atoms of a molecule or ion to be gradually changed to any other arrangement in infinitely many ways, by moving each atom along an appropriate path. However, changes in the positions of atoms will generally change the internal energy of a molecule, which is determined by the angles between bonds in each atom and by the distances between atoms (whether they are bonded or not).
A conformational isomer is an arrangement of the atoms of the molecule or ion for which the internal energy is a local minimum; that is, an arrangement such that any small changes in the positions of the atoms will increase the internal energy, and hence result in forces that tend to push the atoms back to the original positions. Changing the shape of the molecule from such an energy minimum  to another energy minimum  will therefore require going through configurations that have higher energy than  and . That is, a conformation isomer is separated from any other isomer by an energy barrier: the amount that must be temporarily added to the internal energy of the molecule in order to go through all the intermediate conformations along the "easiest" path (the one that minimizes that amount).
A classic example of conformational isomerism is cyclohexane.  Alkanes generally have minimum energy when the  angles are close to 110 degrees. Conformations of the cyclohexane molecule with all six carbon atoms on the same plane have a higher energy, because some or all the  angles must be far from that value (120 degrees for a regular hexagon). Thus the conformations which are local energy minima have the ring twisted in space, according to one of two patterns known as chair (with the carbons alternately above and below their mean plane) and boat (with two opposite carbons above the plane, and the other four below it).
If the energy barrier between two conformational isomers is low enough, it may be overcome by the random inputs of thermal energy that the molecule gets from interactions with the environment or from its own vibrations.   In that case, the two isomers may as well be considered a single isomer, depending on the temperature and the context.  For example, the two conformations of cyclohexane convert to each other quite rapidly at room temperature (in the liquid state), so that they are usually treated as a single isomer in chemistry.
In some cases, the barrier can be crossed by quantum tunneling of the atoms themselves. This last phenomenon prevents the separation of stereoisomers of fluorochloroamine  or hydrogen peroxide , because the two conformations with minimum energy interconvert in a few picoseconds even at very low temperatures.
Conversely, the energy barrier may be so high that the easiest way to overcome it would require temporarily breaking and then reforming one or more bonds of the molecule.  In that case, the two isomers usually are stable enough to be isolated and treated as distinct substances.  These isomers are then said to be different configurational isomers or "configurations" of the molecule, not just two different conformations. (However, one should be aware that the terms "conformation" and "configuration" are largely synonymous outside of chemistry, and their distinction may be controversial even among chemists.)
Interactions with other molecules of the same or different compounds (for example, through hydrogen bonds) can significantly change the energy of conformations of a molecule. Therefore, the possible isomers of a compound in solution or in its liquid and solid phases many be very different from those of an isolated molecule in vacuum. Even in the gas phase, some compounds like acetic acid will exist mostly in the form of dimers or larger groups of molecules, whose configurations may be different from those of the isolated molecule.