An intoxication write-up means the inmate tested positive for alcohol or another intoxicant through a breathalyzer or urinalysis. It is distinct from a possession charge, which would mean staff found the substance itself. Intoxication means the substance made it into the body and showed up in testing.
How it gets there is where inmates demonstrate a level of creativity that never stops surprising people. The most common source is pruno, also called prison wine or toilet wine depending on who you ask. It is a fermented concoction made from whatever is available, typically fruit, fruit juice, bread for yeast, sugar, and water, all sealed up in a bag or container and left to ferment for several days. It smells terrible, tastes worse, and has an unpredictable alcohol content. It is also absolutely everywhere in correctional facilities, bought and sold within the informal prison economy.
Other intoxicants find their way in through visiting room contact, corrupted staff, thrown packages over fences, and other creative smuggling methods. Synthetic marijuana and other substances have become increasingly common because they are harder to detect in standard drug tests.
The consequences for an intoxication write-up are serious. Loss of privileges, a SHU stay, potential good time forfeiture, and in some cases new criminal charges depending on the substance and the jurisdiction. It is not worth what the pruno costs.