Transient response is the measure of a microphones ability to capture a very sudden sound “attack”. The limitations of a microphones transient response are dependent on the diaphragm. Smaller diaphragms have a better transient response due simply to the fact that the they are smaller and can react quicker than a larger one. Also, ribbon and condenser microphones will generally have a better transient response than dynamic microphones regardless of size due to the distinct differences in element design, as explained in Types of Microphones.
Transient response is not as important in vocal applications as it is in capturing percussive sounds such as drums, piano, and plucked strings. The “attack” of the sound these instruments create is an intrigal part on the instruments tonality, so the ability to capture these transients of “attacks” is important.
(In the image, the transient response on the top is from a condenser, and the transient response on the bottom is from a dynamic.)