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Place a normal cauldron above a fire-themed block, such as lava, a campfire, fire on netherack, soul campfire, soul fire, etc.
Fill the cauldron with water.
Clicking a properly setup cauldron with a brewing ingredient will add one of that ingredient to the cauldron. Using the correct ingredients in the correct amounts is a factor in how sucessful your brew will be.
Add your desired ingredients to the cauldron quickly together, then set a timer or judge by feel how much time has passed. How long you cook each brew is another factor in how successful your brew will be.
Once you are satisfied with how much time has passed, right click the cauldron with empty bottles to remove the brew from the cauldron. You can remove three bottles worth of a brew from each cauldron batch.
Brewing stands do play a small role in this unique brewing system, though it is only a small piece of the larger puzzle of brewing.
To get a brewing stand ready to distill some brews, simply place one glowstone dust in the top slot of the brewing stand. This glowstone dust will never be consumed, but is necessary for the distillation process.
Place up to three brews in the bottom slots of the brewing stand. The normal brewing stand animation will happen, after which, each bottle will be labelled as having been distilled once. Leave bottles in for longer to distill multiple times. The number of distillations performed is yet another factor in how successful your brew will be, if distillation is required for your particular recipe.
Seen here are the two sizes of Brewery Barrel, small and large. They are created by assembling the blocks shown, in exactly the same way.
Small barrel: 8 stairs arranged as shown, with a sign on front containing "barrel".
Large barrel: 16 stairs, 20 planks, and 5 fences arranged as shown, with a sign at the top center block containing "barrel".
For the large barrel, the fences ARE required for legs and for the spout. The small barrel requires no legs or spout.
Clicking a properly created barrel will open a small storage space similar to a chest. Place in-progress brews in this storage space to let the aging process commence. Remove the brew after your desired time. Getting the aging period just right is the final factor in how fine your brew is once it is completed, if aging is required for your recipe.