AUTO POTION MAKER

Step by step

Brewing Station Tutorial
First place a chest with a hopper feeding into the back of it. This is where the potions will be delivered after brewing.
To attach the hopper to the back of the chest, hold down "SHIFT".
Brewing Station Tutorial
Place a Brewing Stand on top of the Hopper.
Brewing Station Tutorial
Place a Wooden Button on a solid block next to the Brewing Stand.
It needs to be a Wooden Button, not a Stone one.
Brewing Station Tutorial
Go around the back and place a Redstone Torch on the block with the button, and another block with Redstone Dust as shown.
The block with the Redstone Dust on top will be powered, and this will inactivate the Hopper underneath the Brewing Stand. When the button is pressed, the power will go off and the Hopper will suck the potions out of the Brewing Stand.
Brewing Station Tutorial
Now place two more Hoppers feeding into the Brewing Stand: one from the top and one from the front.
Brewing Station Tutorial
Place a Chest on top of the front Hopper. This is where Water Bottles will be stored.
Brewing Station Tutorial
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.
Brewing Station Tutorial
Place a dropper feeding into each of these hoppers. You will also need a solid block behind each dispenser.
Brewing Station Tutorial
Top view: Place some more blocks to provide a platform for the redstone wiring.
Brewing Station Tutorial
Place Redstone Dust and 3 Repeaters as shown. One of the repeaters will "lock".
This arrangement of Redstone Repeaters functions as an Edge Detector. We use it here to convert the long pulse from the Wooden Button into a short pulse.
Brewing Station Tutorial
The Finished Brewing Station - Front View.
Brewing Station Tutorial
The Finished Brewing Station - Back View.
Brewing Station Tutorial
The Finished Brewing Station - Side View.
Brewing Station Tutorial
The Finished Brewing Station - Side View.

Operation

Loading the ingredients

Brewing Station Tutorial
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.
Brewing Station Tutorial
You can access the brewing stand to see if brewing has finished.

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