Cryogenics is the scientific study or production of extremely low temperatures (below –150 °C, –238 °F or 123 K), whereas cryonics is the low-temperature preservation of humans quickly after the cessation of heartbeart in an anticipation of future survival.
Low-temperature science is very important to various domains of technology: during WWII, it was found that metals cooled to extremely low temperatures were more durable in the field, a process called cryogenic hardening. Liquid nitrogen was then, as it is now, the most commonly used cryogenic agent, as it has a temperature below −320 °F (−196 °C, 77 K). When even lower temperatures are required, liquid helium is used, with a temperature below 3 K.
Cryogenics has many practical applications: in preserving food products or biological samples, blocking water flow in pipes so they can be worked on, in areas where a tap is inaccessible, a coolant for extremely sensitive sensors or overclocked computers, cooling medium for machining certain alloys, and cryotherapy such as removal of warts. Cryonics is also an application of cryogenics, but the two are certainly not the same.
Cryonics is popular in the futurist community as a method of preservation for possible future revival. In common wisdom, it used to be when the heart stopped, a person was defined as dead. But modern medicine allows the revival of those with stopped hearts, so the definition of death has generally been redefined as the cessation of brain activity. Cryonics advocates take this a step further, stating that if the pattern of our neural interconnections (which encodes our personality, memories, emotions, everything) are frozen at extremely low temperatures, then they will not degrade, and the person should not be defined as "dead" per se. Given sufficiently advanced technology, the patient could be warmed up to room temperature and their metabolism rebooted.
There are examples of this in nature: certain frogs can freeze solid during the winter and come back to life during the summer. The cryonics process has been developed such that expanding ice crystals are not an issue: a process called vitrification completely avoids the creation of ice, using flash-freezing, the brain becomes frozen in a plastic-like substance.
Whether or not cryonics ultimately works remains to be seen. But for now: make sure you know the difference between cryonics and cryogenics.