So you want to brew your own beer!  Good, here is a list of some equipment you might need.  This list is in no way comprehensive.  That said most home brew supply shops will have a basic equipment kit for sale that will get you started.

Grain Mill - A grain mill is used to grind/crush the grain in preparation to be mashed.  The two main types of mill available are the Corona-style mill, and the roller mill.  When making beer you do not want to turn the grain in to flour, just crack the husk.

Mash Tun - A mash tun is needed to mash the crushed grains which will extract the sugars used by the yeast.  A mash tun can be any container large enough to hold the grain and water used to mash the grain.  Many home brewers use either a converted water cooler (igloo), stainless steel pot or converted keg for this step.  You can even use primary fermentation bucket for this, just be sure that you can hold a constant temperature for about an hour and have some sort of valve attached to release the wort when done.

Kettle - A kettle will be used to boil the wort.  For extract brewing a 3 to 4 gallon stock pot will work, because you do not have to boil the full 5 gallons.  If you are able to do a full boil then you will need a bigger kettle that fits your needs.  It has been debated on many home brew forums as to the best material to use in a kettle (stainless steel or aluminum).  I don't think it matters to much, but would go with stainless steel if you have the choice.

Wort Chiller - This is a device used to chill the wort after you are done boiling.  Many people use an immersion wort chiller which is a coil of either copper or stainless steel pipe that is put in the wort and cold water run through to help cool the wort.  You can also use an ice bath for this step, but this will take longer.

Primary Fermenter - This is important!  This one piece of equipment will become the main container where the yeast converts the sugars from the wort into ethyl alcohol and carbon dioxide gas.  Most home brewers at some point have used a food grade plastic bucket as a primary and many still do like me.  These can be anywhere from 6.5 to 7.9 gallons to ferment a 5 gallon batch of beer.  Many home brewers also use 6 gallon glass carboys as a primary fermenter because it eliminates the use of plastics and lets them see the fermentation process.  Use a airlock to keep air from getting in, while letting gas to escape. Large glass carboys full of beer can be dangerous if you drop one, so if you take this route be careful.  You can also get plastic carboys from Better Bottle.

Secondary Fermenter - A secondary fermenter is used for aging and clarification of the beer.  Almost all home brewers that do secondary fermentation use some sort of carboy (glass or Better Bottle).  Most use a 5 gallon carboy to eliminate head space, which limits the amount of air (oxygen) in the container.  You can also use a plastic bucket for this step.  This step is not necessary to brew a great beer, but will let the beer age without setting on top of the yeast bed and help get the beer as clear as possible.  An airlock is needed for the secondary also.

Bottling Bucket - This is a bucket with a spigot used to bottle the beer.  You transfer the beer from the fermenter into this bucket adding the correct amount of priming sugar using an auto siphon.  Then you would use a bottle filler to fill the bottles and a capper to cap them. 

Keg/Bottles - Once the beer is done fermenting you are going to need to transfer it in to a container to be carbonated.  This can be beer bottles (new or used) or a keg.  If you decide to use beer bottles then you will also need bottle caps and a bottle capper.  If you go the keg route then you will need a CO2 tank and fittings.