Items needed for brewing beer

Mash tun (make your own:  http://www.youtube.com/watch?v=ftY_HZAFJFc)
Food grade 6 gal+ fermenting bucket with lid/grommeted hole for airlock
Airlock(s)
Siphon or racking cane
7+gal boil pot
Secondary fermenter (usually a carboy) with lid/cork with airlock hole
Quick-read thermometer
Stir spoon/paddle
Hydrometer(s)
Star-san sanitizer
Wort chiller (make your own: http://www.youtube.com/watch?v=YTNB9zBIcvQ
Burner (eventually)
 

Nice things to have (approximate relative order of importance):

Digital scale
Funnel
Sponges/rags/towels
Nylon straining sack(s)
Extra tubing
Carboy handle and/or carry straps
Carboy brush
Carboy drying stand
Homebrew software (e.g. Brewsmith, BeerTools Pro)
Wine thief
Refractometer
Yeast starter kit (stirplate, magnets, and flask)
Fermenter lid opener
Hop grinder
Grain crusher (cereal killer)
Kegging system
 

Bottling Items:

Bottle sanitizer pump
Priming sugar
Bottles
Capper
Caps
Bottle filler
Bottle drying rack
Label paper (optional)
 

Supplies for brew day:

Propane tank
Grains (2-row or wheat + specialty grains)
Hops
Yeast
Augments (e.g. ginger, coffee, cocoa, vanilla beans, Turbinado sugar, honey, molasses, etc.)

 #beer