Correlation spectroscopy
Correlation spectroscopy is one of several types of two-dimensional nuclear magnetic resonance (NMR) spectroscopy.  Other types of two-dimensional NMR include J-spectroscopy, exchange spectroscopy (EXSY), and Nuclear Overhauser effect spectroscopy (NOESY).  Two-dimensional NMR spectra provide more information about a molecule than one-dimensional NMR spectra and are especially useful in determining the structure of a molecule, particularly for molecules that are too complicated to work with using one-dimensional NMR.  The first two-dimensional experiment, COSY, was proposed by Jean Jeener, a professor at Universite Libre de Bruxelles, in 1971.  This experiment was later implemented by Walter P. Aue, Enrico Bartholdi and Richard R. Ernst, who published their work in 1976.

# Principles
A two-dimensional NMR experiment involves a series of one-dimensional experiments.  Each experiment consists of a sequence of radio frequency pulses with delay periods in between them.  It is the timing, frequencies, and intensities of these pulses that distinguish different NMR experiments from one another.  During some of the delays, the nuclear spins are allowed to freely precess (rotate) for a determined length of time known as the evolution time.  The frequencies of the nuclei are detected after the final pulse.  By incrementing the evolution time in successive experiments, a two-dimensional data set is generated from a series of one-dimensional experiments.
An example of a two-dimension NMR experiment is the homonuclear correlation spectroscopy (COSY) sequence, which consists of a pulse (p1) followed by an evolution time (t1) followed by a second pulse (p2) followed by a measurement time (t2).  A computer is used to compile the spectra as a function of the evolution time (t1).  Finally, the Fourier transform is used to convert the time-dependent signals into a two-dimensional spectrum.
The two-dimensional spectrum that results from the COSY experiment shows the frequencies for a single isotope (usually hydrogen, 1H) along both axes.  (Techniques have also been devised for generating heteronuclear correlation spectra, in which the two axes correspond to different isotopes, such as 13C and 1H.)  The intensities of the peaks in the spectrum can be represented using a third dimension.  More commonly, intensity is indicated using contours or different colors.  The spectrum is interpreted starting from the diagonal, which consists of a series of peaks.  The peaks that appear off of the diagonal are called cross-peaks.  The cross-peaks are symmetrical (both above and below) along the diagonal and indicate which hydrogen atoms are spin-spin coupled to each other.  One can determine which atoms are connected to one another by only a few chemical bonds by matching the center of a cross-peak with the center of each of two corresponding diagonal peaks.   The peaks on the diagonal when matched with cross-peaks are coupled to each other.
For example:  a CH3CH2COCH3 molecule 2-butanone would show three peaks on the diagonal, due to the three distinct hydrogen groups.  By drawing a line straight down from a cross-peak to the point on the diagonal directly above or below it, and then drawing a line from the cross-peak directly across to another peak on the diagonal, one can determine which peaks are coupled.  This is done in such a way that the lines from the cross-peak form a 90 angle between the two peaks on the diagonal. The matching peaks, as determined by using the cross-peaks, indicate which hydrogens are coupled, giving a clearer understanding of the structure of the molecule under examination.
To the right is an example of a COSY NMR spectrum of progesterone in DMSO-d6.  The spectrum that appears along both the x- and y-axes is a regular one dimensional 1H NMR spectrum.  The COSY is read along the diagonal - where the bulk of the peaks appear.  Cross-peaks appear symmetrically above and below the diagonal.

# COSY NMR
COSY-90 is the most common COSY experiment.  In COSY-90, the sample is irradiated with a radio frequency pulse, p1, which tilts the nuclear spin by 90. After p1, the sample is allowed to freely precess during an evolution period (t1). A second 90 pulse, p2, is then applied, after which the experimental data are acquired. This is done repeatedly using a series of different evolution periods (t1). At the conclusion of data acquisition the data is Fourier transformed in each dimension to generate the two dimensional spectrum. It is only because the evolution period is varied that cross-peaks appear in the spectrum.
Cross-peaks result from a phenomenon called magnetization transfer. Depending on the experiment, this transfer can be achieved through space or bonds, or even through chemical or physical means.  In COSY, magnetization transfer occurs through the bonds.
Another member of the COSY family is COSY-45.  In COSY-45 a 45 pulse is used instead of a 90 pulse for the first pulse, p1.  The advantage of a COSY-45 is that the diagonal-peaks are less pronounced, making it simpler to match cross-peaks near the diagonal in a large molecule.  Additionally, the relative signs of the coupling constants can be elucidated from a COSY-45 spectrum.  This is not possible using COSY-90.  Overall, the COSY-45 offers a cleaner spectrum while the COSY-90 is more sensitive.  Related COSY techniques include double quantum filtered COSY and multiple quantum filtered COSY.
COSY NMR has useful applications.  Organic chemists often use COSY to elucidate structural data on molecules that are not satisfactorily represented in a one-dimensional NMR spectrum.  Using cross-peaks, along with the diagonal spectrum, one can often discover much about the structure of an unknown molecule.

# NOESY
In NOESY, the Nuclear Overhauser effect (nOe) between nuclear spins is used to establish the correlations.
Hence the cross-peaks in the resulting two-dimensional spectrum connect resonances from spins that are spatially close.  NOESY spectra from large biomolecules can often be assigned using Sequential Walking.
The NOESY experiment can also be performed in a one-dimensional fashion by pre-selecting individual resonances.  This only reveals which peaks have measurable nOe's to the resonance of interest but obviously takes much less time than the full 2D experiment.