I've heard the term brumation used when talking about various reptiles. I haven't heard the term used with any animal besides reptiles, so I'm assuming it's something specific to reptiles. So what is brumation?
You explain brumation quite nicely already in your own answer. I just wanted to expand on it a little and also contrast with hibernation, since that's generally the more well-known term.
Hibernation and brumation look very similar from the outside and indeed serve a similar purpose (getting through the colder/harsher months) but the way the mammal or reptile body undergoes it is slightly different. Mammals build up a fat store before going into hibernation and then go into a deep sleep from which they do not awaken until springtime, living off that fat during their sleep. Most mammals that do hibernate are from very cold climates, where - as you pointed out - food may not be available during the winter months. Their heart rate, breathing and metabolism slows to the barest minimum that would keep them alive.
Reptiles, on the other hand, do not enter nearly such a drastic hibernation period. Not all reptiles brumate, but those that do usually eat slightly more in the months leading up to brumation, and during the colder months they can be lethargic and eat less if at all. Unlike mammals, they do still need to drink water, and do not sleep the months away like mammals do (though they will sleep more, and be more lethargic in their waking hours).
Many breeders will lower the temperature of the vivarium to induce brumation, because breeding will often not be successful without a period of brumation beforehand. With a constant temperature, reptiles may not brumate - or they may respond to other signals like lengthening nights and brumate anyway. If you choose to lower the temperature for your reptile and continue to offer it food, you must make sure that it has access to a heat source. Allowing access to food but not heat can be dangerous because reptiles rely on external heat sources to digest their food. If food is eaten but not digested, it can literally rot in the stomach.
Finally, sometimes the word hibernation is used for reptiles as well, because the processes are so similar and the general public is more familiar with the word hibernation than brumation.
Brumation is a form of hibernation that certain species of reptiles will go through. The reason that there are different terms for the two forms, is because the two have different metabolic processes.
The best way I can describe it is that hibernation is something that warm-blooded animals have chosen to do through the process of evolution, because it saves them from trying to survive during months where there might be little to no food.
For example: Insects that live for multiple years will hibernate because the plants that they feed on are under snow and ice, which in turn makes animals like bats and frogs that feed solely on insects hibernate because it would be too inefficient to spend energy finding the insect)
Brumation is something that cold-blooded animals are forced to do, because their bodies are so directly linked to the outside temperature. That is why reptiles that come from tropical climates, where it stays warm pretty consistently, will not brumate. [Note: There are some reptiles that live in consistently cooler regions (like mountain areas) that don't necessarily brumate, because of how their species evolved to live in the colder climate]
Because reptiles need certain temperatures to absorb nutrients from their food, the colder temperatures means they cannot gain energy even if they did eat. So, their bodies go into the state of sleep similar to hibernation to keep from starving to death. Most reptiles will "wake up" every few days as they still need to stay hydrated.