A large part of brewing any sort of fermented beverage is cleaning! It's imperative that your cleaning products are free of rinsing agents, dyes, harsh chemicals, perfumes, or bleach. As such, I recommend using OxiClean Free (or a market analog) or PBW. If you must use soap, make sure that it too is free of any additives, and be sure to rinse your gear extra well.
Cleaning your equipment is half the battle, and sanitizing it is the other half! Its crucial that you practice strict sanitation practices when brewing fermented beverages, as they can be highly susceptible to infection (especially before the fermentation has taken place). I personally recommend Star San, which has been developed for brewing applications. You can fill a spray bottle with Star San and keep it on hand for quick sanitizing applications.
For your primary fermenter, I personally recommend using a 6 gallon high-density polyethylene (HDPE) bucket. These plastics are marked as a #2 recyclable and are food-grade material. The reason I recommend these over a glass carboy for your primary fermenter is that they are cheaper to purchase, easier to fill after brewing, and much easier to clean after fermentation is complete.
Beer and Wine Triple Scale Hydrometer:
Although one could argue that a hydrometer is not essential to the process of brewing fermented beverages, it is essential if you wish to know the alcohol percentage of your beverage. A hydrometer measures the density of liquids using a measurement called specific gravity. In the case of alcoholic beverages, the more sugar your beverage has in it, the higher the density, and thus the higher the specific gravity. By taking a gravity measurement before and after fermentation takes place, in which the sugars are converted into alcohol (and thus the gravity drops), one can ascertain the amount of sugar that has been converted into alcohol. Not only does this allow one to determine the overall alcohol content of your finished beverage, but also its residual sweetness.
In order to boil your wort (i.e., the sweet fermentable liquid that will become your beer), you need to have a decent-sized pot. If you will be doing 5-gallon brews, which is the standard volume for home brewing, I would encourage you to get an 8-gallon pot, or larger. This is because the wort has the propensity to foam a great deal at the beginning of a boil, and if you have a larger pot, you are much less likely to get a boil-over (which is a frustrating mess, to say the least). You want to use only either aluminum or stainless steel, and make sure the pot has decently thick walls (that way your wort will be less likely to scorch). If you wish to use aluminum (which I personally use), make sure you season it prior to your first boil. Just Google "seasoning aluminum stock pot" and you'll find instructions on how to do this.
You will need a spoon for stirring your boiling wort and hops, as well as many other times throughout the brew process. I would recommend getting a spoon with at least a 21" long handle, and I would recommend against using plastic or wooden spoons.
It is very important to know the temperature of your wort at various times throughout the brew process. I'd recommend getting a long-stemmed milk thermometer (like the ones used by baristas), which has a range of at least 60°F (15.6°C) to 190°F (87.8°C).
Airlock and Stopper:
In order for your fermenting wort (or must, if you are brewing wine or cider) to remain free of contaminants, you need an airlock and stopper. The airlock allows the CO2 being generated by the fermentation process to escape while not allowing any other particulates into your fermenter. Fill your airlock with sanitizing solution, as reverse pressure can cause some of the solution to be sucked into your fermenter (this does happen on occasion).
Auto-Siphon and Hose:
Having an auto-siphon is not essential, but having a siphon is, and this is why I encourage everyone to just pay a little extra and get the auto-siphon at the beginning. You will use a siphon during bottling and racking. Auto-siphons are much easier to use and less prone to infection than your standard siphon. Make sure to get a hose that is about 5 feet (1.5 meters) long.
There are many styles of cappers, so just choose one that fist you best. This device does exactly what it sounds like it does: it crimps bottle caps onto your bottles.
12 or 22 oz Bottles:
You can buy bottles at a brew store, online, or even on Craigslist (there are many awesome home brewing finds on Craigslist). Also, you can get bottles from recycle bins or have your friends save them. Make sure you clean them out with the cleaning agents discussed above and then sanitize them thoroughly. If you decide to use old beer bottles that have labels on them, you can remove the labels with just a bit of work. I have written a post about how to completely remove labels, which you can find here.
Bottle caps are inexpensive and essential (unless you are using swing-top Grolsch style bottles). You need to buy new caps as you cannot reuse old bottle caps. There are regular caps and CO2 absorbing caps, which cost a bit extra. You do not need to buy CO2 absorbing caps, no matter what the dealer tells you, unless you are bottling from a keg (which you most likely will not be doing as a new home brewer).
Non-Essential (but EXTREMELY helpful) Equipment:
"How to Brew," by John Palmer:
For anyone who likes to brew beer, whether they are a beginner or an avid brewer, this book will expand their understanding of the science behind the process and help them hone their craft. I recommend buying his book, as it is full of up-to-date information, however you can also view the first edition of his book on his website for free.
There are many kinds of wort chillers out there, but I believe the copper-coil style (pictured here) is the least expensive and most easy-to-use option. It is imperitive that you cool your boiled wort as quickly as possible in order for the cold break to occur. The cold break is a process of "shocking" proteins in the wort so they congeal and settle at the bottom of your brew pot. Either you use a wort chiller, or you can use an ice bath in your bathtub. If you use an ice bath, you will need a substantial amount of ice, and the process can be slow. The wort chiller is a much easier and efficient option.
Large Mesh Strainer:
When pouring your wort from your boil pot into your primary fermenter, you want to keep the hop particulates and protean lees from making it in there as well. You can technically just pour the wort very carefully, while using your brew spoon to hold back the hops, but a large strainer makes the process much easier.
High Pressure Propane Burner:
It is really important that you are able to boil your wort in a quick and efficient manner. You can do this inside if you have a gas stove, but if you have an electric stove, the process can take hours longer. Even if you have a gas stove, I'd recommend investing in a propane burner, as they are much more efficient and will heat your wort much quicker. Be sure to never use these devices inside.
Nylon Mesh Grain Bag:
Having a grain bag is great if you are doing an extract brew, as it will allow you to steep specialty grains (i.e., specific malts that will give your beer more character) in your wort. Also, you can use it to hold fruit, herbs, or vegetables that you want to steep in your beers and wines. You can buy these mesh bags from brew stores, or you can simply buy a paint strainer bag, which is essentially the same thing item but is often sold at a much cheaper cost.
Filling your beer bottles can be a pain without one of these, which is why I recommend them. You attach this filler to the end of your siphon prior to bottling. A spring-loaded valve at the end of the filler stops the fermented beverage from flowing until you press it into the bottom of a bottle, at which point the valve is opened and the bottle fills. Also, this device substantially cuts down the risk of oxidation, which can happen if you splash too much during bottling. Once your beverage is oxidized, it will take on the flavor of wet cardboard within a few months. Understandably, you really want to avoid this from happening, especially if your beverage is one that you wish to keep bottled for more than a month or two.
Similar to aquarium thermometers, this adhesive thermometer sticks on to your fermenter buckets or carboys and allows the brewer to easily monitor the beverages temperature during fermentation. Monitoring and regulating temperature throughout the fermentation process is essential for high quality brews. You can also open your fermenter and take temperature readings with a thermometer, but this greatly heightens the chances of infection. With the fermometer, you can monitor the temperature accurately enough while never having to open your fermenter.
This device allows you to draw samples from your fermented beverage for gravity measurements and quality checks without disturbing it. Furthermore, the tube is thick enough that you can take gravity measurements directly in the wine thief.