From a given number of atoms, you can usually construct multiple molecules (isomers), so we shall just concentrate on the case where the bond connectivity is the same.
The terms you are looking for are conformation and configuration. The latter is most usually related to double bonds and E/Z isomerism. If we take but-2-ene, for example, it can exist in 2 forms, E and Z. Both are stable and have the same atom connectivity. However, they cannot interconvert between each other - the activation energy is too high.
Two conformations of the same molecule, however, can interconvert between each other (at least at room temperature) and they are both stable (i.e., potential energy minima). A classic case of a molecule is butane, whose conformations can be seen in the picture below.
The anti- and gauche- conformations are both 'butane' and both are stable (even though the anti- is more stable), but have a different 3D arrangement of atoms.
As a sidenote, these can be distinguished at a low enough temperature, at which they cannot interconvert.
Another famous molecule with various conformations is cyclohaxane, with the two stable ones being the chair and the much less common twist-boat.