Under normal conditions, the recoil of a free nucleus when it emits a gamma-ray subtly reduces the gamma-ray energy. The gamma-ray then does not have enough energy to excite another nucleus to undergo the same transition, and so what is known as resonant absorption of the gamma-ray is suppressed. In practice, thermal Doppler broadening of the gamma-ray emission spectrum (and the absorption spectrum) means there is a small chance of resonant absorption. As a sample is cooled, the thermal broadening is reduced, and so should resonant absorption. In essence, the Mossbauer effect is a demonstration the resonant absorption increases at low temperatures, contrary to expectation.
Mossbauer explained this by considering the gamma-ray emission and absorption as essentially recoiless at low temperatures, as the nuclei are frozen into a lattice and the whole lattice recoils only very little compared with how one nucleus recoils. Also, since the nuclei are all in low vibrational states due to the reduced temperatures, the emission and absorption line profiles are very narrow.
The Mossbauer resonance is thus very narrow, and only very small changes in the nuclear energy levels are needed to destroy it. Thus the Mossbauer effect can be used to create very sensitive spectrometers, which can be used for a variety of applications.
For more information, check out the Mossbauer Information Exchange.