If you want to brew with the least amount of equipment you would need at least the following on your first brew day:

* Sanitizer - Cleanliness is incredibly important to the process. I prefer Starsan. Works great and is no rinse.

* Brew Pot - Any standard stock pot will work. However, a bigger pot will make your beer turn out better. (Although you can get away with just using a standard 3 gallon one and topping off with bottled, distilled water.)

* Fermentor - You will need something to keep the wort in as it becomes beer. It will need to be oversized by ~1 gallon to provide room for the krausen (foam that forms as yeast do their thing). Buckets with lids are the standard here.

* Airlock and grommet - You need a way to let out CO2 without letting in air. An airlock does this perfectly and a grommet keeps it airtight.

What you need for bottling:

* Bottles - a standard homebrew batch is 5 gallons so you will need 48-52 bottles. Easiest way is to just save any poptop beer bottles and use those.

* Caps - something to close the bottles!

* Capper - to put the caps on

* Racking Cane and Hosing - to pull beer out of fermentor.

Like I said this is just your absolute minimum list. To make things easier I would recommend-

* Bottle Filler - A gravity/spring loaded rod that only dispenses beer when the bottom is pressed against the inside of a bottle.

* A separate bottling bucket - Allows a more even mix of priming sugar and prevents trub (beer gunk) from getting into bottles.

* Thermometer and Hydrometer - Pretty much consider these essentials. Allow you to control your brew more accurately.

* Secondary Fermentor - This is a pretty contested topic in the world of homebrew. Some people say that they are just an additional infection / oxidation risk. Others (like me) prefer to use them and feel they allow you to make clearer, better beer. A carboy is the standard option here.