What equipment do you feel is essential for brewing? Here is my (very obvious) list.
A kettle (stock pot) of at least 5 gallons (20 quarts) for boiling your wort. A 5 gallon kettle is just big enough to do partial boils on your stove top.A 7 gallon kettle would be better so you could do full boils, but you might want to look into getting a turkey frier set up because your home stove is probably not adequate to reach a vigorous boil. It should be mentioned that Stainless Steel or Enameled Steel is preferred over Aluminum because some (myself not included) believe that Aluminum can leech toxins into the wort. I use Stainless Steel but that doesn't mean you have to.
A fermentation vessel. This could be an ale pale, a carboy, better bottle, etc. As long as it is food grade you should be good. This should be at least 6.5 gallons to allow head room for the krausen. I prefer the models with a spigot so that I don't have to worry about siphoning. The downfall with the spigot however is that you can potentially end up with more trub in your bottling bucket/secondary fermenter than if you used a siphon. If you go the bucket route you will need a lid that is drilled out to accept a stopper to use with the airlock. If you go the carboy/better bottle route either a stopper or carboy cap. I like the carboy caps personally. They accept an airlock, or can be setup with a hose to be used as a blow off tube.
A bottling bucket. This is basically just a fermenter that you don't use a lid on. Your beer won't be in this bucket very long. I don't have a keg setup, but i would think that if you are naturally carbing your beer in the keg you would still want to use a bottling bucket to mix the priming sugar in before you transfer to the keg. This bucket should have a spigot to facilitate bottling and transferring. Along with the bottling bucket, you will want a bottling wand to make bottling that much easier. Again if you are kegging, the bottling wand is not needed.
An airlock. The airlock allows co2 to escape from the fermenting wort while preventing anything from entering the fermenter. The airlock is filled with a liquid. This could be plain water, sanitizer water, vodka. Never use anything that you would not put into your beer, ie no isopropyl acohol (rubbing alcohol). You can also rig a blow off tube to your fermenter for those extra vigorous fermentations. A blow off tube is just a length of hose connected to the fermenter with the other end in a small container of sanitizer solution. It acts just like a big airlock only if you get krausen in it, it (hopefully) won't clog up resulting in your fermenter blowing up.
A thermometer. Without this how can you tell when it is safe to pitch yeast or how hot your water is if you are using steeping grains?
A length (about 4-6 feet) of food grade vinyl tubing. This is used for everything related to transfering wort.
Some bottles. roughly 48, 12 oz bottles for a 5 gal batch or 24, 22 oz bottles. I prefer the 22 oz bottles because there are less to fill and will lets be honest, is 12 oz really a single serving? You can of course use whatever size you want, just make sure you have enough for the batch. Now if you keg you wont "need" bottles, but if you wanna send a sixer home with a buddy you will probably wanna keep some on hand.
A bottle capper and caps. Can't bottle if you can't cap the bottles. Pretty self explanatory.
Now for the honorable mentions. These are items that you don't technically need, but are definitely wanted.
A triple scale hydrometer. This is used to get the specific gravity of you wort and final beer. A hydrometer will allow you to determine when fermentation is finished. You will also be able to determine the ABV of your brew by taking your Original Gravity and Final Gravity and doing some calculations.
A scale. So you can weigh out your ingredients if you need to. Like if you are doing starters, late additions, etc.
Hop bags/ Muslin pouches. If you are like me, you don't want to have to strain out your wort. So hop bags (grain bags if using steeping grains) can help with this.
Various Glassware. No one wants to drink a frosty cold beer out of a regular glass. I mean.... you can. But it looks so much cooler if you have some nice beer mugs, pint glasses, pilsner glasses, etc.
I think that basically covers it. Feel free to add anything I may have forgotten. I'm typing this after having a few homebrews so i am sure I left something out.