Two-dimensional infrared (2D IR) vibrational spectroscopy is an experimental tool for investigating molecular dynamics in solution on a picosecond time scale. We present experimental and theoretical methods for obtaining a 2D IR correlation spectrum and modeling the underlying microscopic information. Fourier transform 2D spectra are obtained from heterodyne-detected third-order nonlinear signals using a sequence of broad bandwidth femtosecond IR pulses. A 2D IR correlation spectrum with absorptive line shapes results from the addition of 2D rephasing and nonrephasing spectra, which sample conjugate frequencies during the initial evolution time period. The 2D IR spectrum contains peaks with different positions, signs, amplitudes, and line shapes characterizing the vibrational eigenstates of the system and their interactions with the surrounding bath. The positions of the peaks map the transition frequencies between the ground, singly, and doubly excited states of the system and thus describe the anharmonic vibrational potential. Peak amplitudes reflect the relative magnitudes and orientations of the transition dipole moments in the molecular frame, the electrical anharmonicity of the system, and the vibrational relaxation dynamics. The 2D line shapes are sensitive to the system-bath interactions in solution. We illustrate how 2D IR spectra taken with varying polarization conditions and as a function of a variable waiting time can be used to isolate and quantify these spectroscopic observables. As a model vibrational system, we use the strongly coupled asymmetric and symmetric carbonyl stretches of Rh(CO)(2)C5H7O2 (RDC) dissolved in hexane and chloroform. The polarization-selective 2D IR spectra of RDC in hexane are analyzed in terms of two coupled local coordinates to obtain their mutual orientation and the magnitude of the coupling between them. The 2D line-shape study of RDC in chloroform performed as a function of the waiting period characterizes the system-bath interactions, revealing that the system transition energies fluctuate in a correlated manner.