From AMS Glossary
In general, the time interval required for a system exposed to some discontinuous change of environment to undergo the fraction (1 - e-1), or about 63%, of the total change of state that it would exhibit after an infinitely long time.
For example, a thermometer initially at equilibrium in a bath at temperature T1 will exhibit an exponential change of temperature with time after being suddenly plunged into a bath at temperature T2, theoretically assuming the new temperature T2 only after an infinitely long time. The finite time interval required for the thermometer to undergo a change of amount (T1 - T2)(1 - e-1) is called the thermal relaxation time of the thermometer. Occasionally, the fraction 9/10 is used in place of (1 - e-1), so contexts must always be checked to be certain of the definition employed in a given case. The definition may also change for an underdamped device. The change of state of such a device may oscillate several times while approaching its final value.