How do we define the longitudinal relaxation time and transverse relaxation time?
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In an NMR measurement you rotate the spins 90º to the magnetic field. Over time the spins relax back into line with the magnetic field, and this is the longitudinal relaxation time, $T_1$.
When you first rotate all the spins 90º they are all in line, and they start rotating in the horizontal plane to generate the NMR signal. However the spins won't stay in phase. Random thermal interactions will cause the spin rotation rate to vary randomly and the spins get out of phase with each other. This causes the NMR signal to decay even though the spins are still at 90º to the magnetic field. This process is called the transverse relaxation or $T_2$.
A short $T_2$ is a major pain for solid state NMR since it means you quickly lose the signal but you still have to wait several $T_1$s for your sample to relax before you can repeat the measurement.