The minimium amount of equipment needed is:
3 gallon (12 quart) stock pot (called a kettle), aluminum or stainless steel. If you already have a stock pot with lid, you can use it.
6 gallon plastic bucket or glass carboy fermenter
Airlock and rubber stopper
Cleaning and sanitizing powder or liquid
Sponges or non-abrasive scratch pads
Empty beer bottles (11 x 22 oz or 48 x 12 oz)
Racking cane or bottling bucket
Bottle caps and capper
Very little space is required when starting. A small kitchen counter, stove, sink, and floor space is all that is needed. The fermenter can remain in the kitchen, to be placed in a bedroom, closet, or anywhere out of the way. It will remain there for approximately 2 weeks.
Decide on what you want to brew. What kind of beers do you like? Blonde, amber, stout, IPA, red, porter, or something else? There are several dozen recipe kits available, so finding a style you like should be fairly easy. Expect to spend anywhere from $30 to $50 USD depending on the kit. More complex beers, such as Belgian and stouts typically cost more.
Some canned malt kits will require additional dry malt extract packages to be bought in order to provide the necessary amount of fermentable sugars to create the right amount of alcohol. 7 lbs of exact is the most common. You can either buy multiple cans, or the less expensive method is to buy dry malt extract.
All kits come with a package of dry yeast under the lid. You can use this yeast or substitute another variety.
An existing stock pot, whether stainless steel, aluminum, or blue canning pot will work fine. If you do not have, or wish to use an existing pot, you can buy an inexpensive pot anywhere you desire. Discount store, thrift store, many grocery stores will carry stock pots, home improvement, garage sale, classified ads, or anywhere else you wish to go.
Existing wooden or steel spoons and thermometers can be used. Do not use a meat thermometer, as the temperature range is not adequate for brewing.
Buckets and carboys should be bought from a homebrew supply store. A bucket will be the cheapest route. You can get a bucket for lid for under $20 USD. The reason a homebrew store is recommended is because, even though the kit is for 5 gallons, regular buckets are exactly 5 gallons, and beer must have a few inches of head space. Homebrew buckets are often 6.5 gallons.
Airlock and rubber stopper for homebrew supply store.
Cleaning and sanitizing solution can be bought from a supply store, or you can use unscented cleaners. Do not use hand soap. It is recommended to also get these supplies from a homebrew store because they are food grade. Powder cleaners are very cheap, usually only a few dollars for a large package, and sanitizers cost more, but can often be reused.
Any clean sponge or scratch pad can be used.
Beer bottles can be found in many places. When you buy beer, simply keep the bottle, ask family, friends, neighbors, and bars for extra bottles. Many are thrown out. Do not use twist top as they will not seal properly. You can buy bottles at homebrew stores, but most brewers I have talked to get them from friends, coworkers, or ask a bar.
Bottle capper and caps can be bought at a homebrew supply store. I bought some extras on ebay.
A racking cane and tubing is also bought from a homebrew store. However, if you buy a combination fermenting/bottling bucket, you do not need the racking cane.
A full kit can be bought instead of individual items. Depending on the kit, the most common pieces are the bucket and lid, airlock and rubber stopper, cleaning solution, bottle capper and caps, racking cane and tubing. Some parts, like the kettle and spoons are typically not included with a begineers kit and must be supplied by the brewer. These kits cost between $60 to $90 USD.
If you do not have access to a supply store, many are available online, sometimes with shipping promotions, or look on ebay.