|There is more information available on this subject at Cryonics on the English Wikipedia.|
Though often mistaken for cryogenics, which is merely the study of cold on materials, Cryonics is the science of cryo-preservation, using a mixture of factors to preserve a Human or other organism in stasis for periods of time.
Essentially, extremely low temperatures are created, preserving the human body almost indefinitely. Preserving the human brain is more problematic, but UNSC technology has apparently overcome the obstacles of neural tissue damage and resuscitation, but with tissue damage still being a problem, causing what is colloquially called "Freezer burn." However as stated in Halo: Contact Harvest, the UNSC seems to use certain types of drugs to prevent "Freezer burn."
The UNSC uses cryonic storage pods in long-range warships, storing personnel for long periods to prevent aging during the journey and to preserve life-vital systems, maintaining only skeleton crews until nearing their target zone. A bronchial surfactant is ingested, intended to replace nutrients lost during the journey, but the process is unpleasant and often induces vomiting.
It is unknown whether the Covenant use a version of this technology. Given the dramatically higher speeds their warships can achieve, though, it is unlikely that it is necessary. The Forerunners used a system similar in practice, but using slipspace to store the personnel within the pod instead of merely freezing them. In some cases UNSC marines can be placed in cryostasis to help ration food and oxygen as they don't need it if they are frozen.
Cytoprethaline, a drug used by the UNSC during cryo-sleep, prevents damage to the occupants cell membranes caused by ice crystal formation.