Enantiomers are isomers that are mirror images of each other and diastereoisomers are those that are neither mirror images nor the same molecule. In both cases the molecules have the same atoms connected to each other in the same way. The differences between these isomers are the spatial arrangements of the atoms based on the 3D-bond geometry.
Most molecules are not flat arrangements of atoms. The shape of a molecule depends on the types of atoms and covalent bonds that form. For example, because a carbon atom typically forms four different bonds with other atoms, a molecule with a carbon atom at its center has a tetrahedral, or pyramid-like shape. If all four of the atoms in bonds with the carbon atom are different, it's impossible to superimpose one over the other regardless of how the two isomers are turned.
One way to check if two carbon atoms have the same stereochemistry is to position the diagram of the carbon atom so that the attached atom with the lowest atomic number is facing away from you. If you can turn the two molecules so that the other three atoms are in the same position, then they have the same chemistry and are the same molecule.
This process is complicated for molecules containing more than one carbon atom with four different bonds to other atoms. However, it is still possible to determine if the molecules are the same, enantiomers or diastereoisomers by starting with one carbon atom and working through the entire molecule.