Step by step
First place a chest with a hopper feeding into the back of it. This is where the potions will be delivered after brewing.
Go around the back and place a Redstone Torch on the block with the button, and another block with Redstone Dust as shown.
Now place a row of hoppers all feeding into the top Hopper. You will need between 2 and 5 of these depending on which type of potion you want to make.
Operation
Loading the ingredients
Load up the ingredients. Water Bottles go in the top chest. Potion ingredients go in the droppers, in the correct order. The first ingredient should be closest to the Brewing Stand, the second ingredient next closest, and so on.
When you place the Water Bottles, the first 3 will disappear immediately into the Brewing Stand. The next 5 will disappear into the Hopper. After that, the Water Bottles will fill the chest.
Operation
Push the button. Potions will be dispensed into the bottom chest. A new batch of potions will start brewing. (The first time after building the brewing station, you will just get water, but after that it should work).
Allow the potions time to brew before pushing the button again.
Technical Notes
We use a wooden button rather than a stone button, because a wooden button generates a longer redstone signal. With a stone button, there wouldn't be enough time for all three of the potions to be released from the brewing stand.
However, this long signal causes problems elsewhere, If this signal was used to directly trigger the droppers, they would release two items each (one as the button clicks "on", and the other as the button clicks "off". This is why we need the arrangement of redstone repeaters that acts as an edge detector: a short pulse is emitted only as the button clicks "off".
(Sources)
http://www.minecraft101.net/t/potion-brewer.html
http://minecraft.gamepedia.com/Minecraft_Wiki