in a 2D-NMR-spectrum there is shown an 1D-NMR-Spectrum on each axe (for example: x-axe: 1H, y-axe: 1H or x: 1H, y:13C). Such spectra are useful for structure elucidation of complex molecules, because there you can get information on the couplings of each nucleus (for example, each 1H or each 13C).
In 2D NMR you can see which proton nuclei are coupled to eachother through bonds which would give you an idea of which protons are close to each other . You can also see the same thing but with proton to carbon. That is to say, which protons are attached to which carbons. Those two things above are two separate experiments but allow you to determine connectivity of the molecule.
Another 2D experiment allows you to see which protons are close to each other in 3D space (they don’t have to be close to each other in terms of bonds, they can be many bonds away). This is helpful for determining conformation or stereoisomerism of the molecule.
2D NMR is absolutely necessary for characterization of an unknown molecule or natural product. If you are Doing a reaction that is known chemistry and you know the identity of your starting materials, 1D is usually sufficient to confirm if you have synthesized your correct product or not.
Generally monodimensional spectra give enough info to elucidate the structure of simple compounds on the basis of chemical shift, multiplicity and integrals of signals (the latter may be not applicable in all the experiments). With larger compounds and in the case of overlapping resonances the bidimensional experiments may give more and unequivocal indications.
With a selected set of bidimensional experiments it is also possibile to identify the components of simply mixtures.