The T1 relaxation time (also called spin lattice or longitudinal relaxation time), is a biological parameter that is used in MRIs to distinguish between tissue types. This tissue-specific time constant for protons, is a measure of the time taken to realign with the external magnetic field. The T1 constant will indicate how quickly the spinning nuclei will emit their absorbed RF into the surrounding tissue.
As the high-energy nuclei relax and realign, they emit energy which is recorded to provide information about their environment. The realignment with the magnetic field is termed longitudinal relaxation and the time in milliseconds required for a certain percentage of the tissue nuclei to realign is termed 'Time 1' or T1. Starting from zero magnetization in the z direction, the z magnetization will grow after excitation from zero to a value of about 63% of its final value in a time of T1. This is the basic of T1 weighted images.
The T1 time is a contrast determining tissue parameter. Due to the slow molecular motion of fat nuclei, longitudinal relaxation occurs rather rapidly and longitudinal magnetization is regained quickly. The net magnetic vector realigns with B0 leading to a short T1 time for fat.
Water is not as efficient as fat in T1 recovery due to the high mobility of the water molecules. Water nuclei do not give up their energy to the lattice (surrounding tissue) as quickly as fat, and therefore take longer to regain longitudinal magnetization, resulting in a long T1 time.

See also T1 Weighted Image, T1 Relaxation, T2 Weighted Image, and Magnetic Resonance Imaging MRI.