A lot of equipment is personal preference. Read through the list and feel free to ignore anything that doesn't fit your process. This list is more exhaustive than necessary.
Read about the process of sanitation here.
Spray Bottle - keeps no-rinse sanitizer potent for a longer period of time. Convenient for spot-sanitizing, spraying on your hands, or any small piece of equipment without making up an entire bucket.
5 Gallon Bucket - for when you need 5 gallons of the stuff.
Dirty Bucket - 1-5 gallon bucket for gear that needs to be washed. Alternatively, have a sink nearby.
You can read more about the process of fermentation here.
Bucket
Lids are optional. During active fermentation, the mead will be producing enough CO2 to keep it safe from oxidation. If you have fruit flies, however, you'll want to use a lid.
You need a bigger bucket than your intended batch size to avoid foam overflow.
2 gallon buckets for 1 gallon batches.
6.5 - 10 gallon buckets for 5 gallon batches. Melomels are easiest to ferment with larger buckets.
Carboy
Again, you need a larger carboy than your batch size.
Glass or plastic probably doesn't matter for fermentation.
Will work in a pinch, but a bucket is much easier to work with during the early stages of fermentation when using advanced techniques.
You probably want to decide between glass and plastic.
Glass Carboys or PET Carboys / Better Bottles
Same size as planned batch size to minimize head space.
Remember that you need more than one if you intend to make mead more than once every few months.
Food Safe Buckets
Lowes or Home Depot (in the US) often have 5 gallon food grade buckets which may be used for fermentation.
These are very cheap, and work fine for 3 gallon batches.
Big Mouth Bubblers
Available from online retailers in 6.5 and 5 gallon varieties.
Reports of low production quality for the glass variety (bubbles in the glass, eggshell thin sections).
Reports of poor seals on plastic variety.
Relatively new product (2013-2014), your mileage may vary.
No issue using plastic BMB as fermenters.
Whatever vessels you settle upon, make sure you get a bung and airlock that fits each one!
Three piece airlocks are easy to thoroughly clean, but can suck the water/vodka back into your product if the air pressure changes.
S-Curve airlocks can't suck liquid into your mead, but can suck air into the carboy if there's a significant exterior pressure increase.
Solid bungs are useful for moving carboys around.
Good for all sorts of things. It’s, $1.50. The wider the funnel bottom, the less time you’ll be waiting for the funnel to drain when pouring something through it. The wider the funnel top, the less you will spill.
It is essential in brewing to have some method of measuring the specific gravity (density) of your must and finished mead in order to determine how much sugar remains in a must or mead.
The gold standard typically used by most brewers. Ensure your device is not defective but floating it in distilled water (at a temperature close to the calibration temperature printed on the hydrometer) and ensuring it reads 1.000. Also properly adjust for temperature.
Glass hydrometers, by far the most common, have a reputation for being extremely fragile. It is typically advised to always have two on hand for when you inevitably break one. You will also typically need a test tube capable of holding the hydrometer to use when taking measurements. It is not recommended to attempt to float a hydrometer in a glass carboy. If you are fermenting in a bucket, then it is often easier to simply float the hydrometer in the bucket rather than drawing a sample.
Polycarbonate hydrometers have recently become more affordable. While often slightly more expensive than glass hydrometers, they do not suffer the same fragility of glass. As a result, they are likely the better buy for most homebrewers than a glass hydrometer. One downside over glass is that dissolved CO2 will tend to cling to the side of the hydrometer a bit more.
These are great and convenient devices with a chief downside. They do not accurately measure SG for brewing purposes in the presence of alcohol. You will need to use a formula (or calculator) to convert to a useful number.
It is common for a misleading refractometer measure to be mistaken as a stall.
There are other devices like Tilt Hydrometers, iSpindels, the Anton-Par Easydens, etc. Each will come with proper instructions on how to use and maintain the devices. Follow them and ensure you are calibrating correctly, if needed.
Wine Thief - Simpler is better when it comes to cleaning. Steel turkey basters work great, but be very careful not to drop the metal part into a full glass carboy.
Wine Thief / Test Tube combo - Convenient to use, but has moving parts. Less surface area, but harder to thoroughly clean. A matter of preference.
Thermometers - Again, keep two.
12” Glass - Not mercury, alcohol! Fragile. Keep in its tube.
Carboy Sticker Thermometer - Plenty of folks use these to monitor carboy temperature.
Digital Thermometers - These are expensive and sometimes mis-calibrated. But wow are they convenient. Consider if you get serious, or can double it's use for brewing beer.
Sugar Thermometer - not required, but may be useful for some styles.
Refractometer - Really useful, but more expensive. They require a smaller sample than hydrometers, but also require a very clean sample. Are thrown off by the presence of alcohol, but the true reading can be calculated. Not required.
Gram Scale - Optional. If you want to measure nutrients accurately, it's indispensable, but you can get by without. If you get one, take the following into consideration.
Minimum resolution: .01 grams
Maximum weight: 100 grams is plenty.
Kitchen Scale - Optional. Makes it easy to add honey by weight.
Minimum resolution of 1 oz.
Maximum weight: 15-20 pounds.
Measuring Cup - good for rehydrating yeast, too. Easier to clean than Erlenmeyer flask.
Look for borosilicate glass.
2-4 c volume
You want one with ml markings.
Measuring Spoons (set) - You'll need them for something, even if you have a gram scale.
Metal ones will outlive you.
This is not required equipment. If you do use one, it obviously requires a drill you can attach it to.
Plastic
Works fine.
Limited use life. The stirring wings can break eventually.
Stainless Steel
The metal rod is nicer to use because it doesn't bend when stirring the must.
More expensive.
Longer life, replaceable parts.
Alternatively, stir with a big spoon, or shake the carboy to initially mix honey and water.
Never shake a carboy during fermentation. This can create a geyser.
The autosiphon is a mighty gift to homebrewers.
Autosiphon
A simple pump starts the siphon from the source, creating no need for priming the hose with sanitizer or creating suction.
Available in several sizes and diameters.
Tubing - some may come with your autosiphon. Replace every 6-12 months.
It is common to use tubing slightly smaller than the size intended for the autosiphon (e.g. If 5/16" is recommend, use 1/4"). Heat the end of the hose with hot tap water to fit it over the autosiphon. It may take some careful work, but in the end the seal will be better. Don't count on removing it very often (or ever).
Vinyl tubing is most common and very cheap. It is not suitable for temperatures over 130°F (54°C). Available at homebrew shops and hardware stores, usually priced by the foot/yard/meter.
Silicone tubing is much more expensive and unsuitable for high PSI (check with the manufacturer for specifics), and is easy to damage, but it can withstand much higher temperatures.
For the full article on Packaging, which goes into much greater detail, click here.
Though often ignored until the last minute, packaging is an important consideration.
Bottling bucket - this is a bucket with a spout near the bottom.
You can also just use an autosiphon and fresh, sanitized carboy.
Spring-Loaded Bottling Wand
Makes bottling so much easier.
Cheap, should last for years.
Get one that fits onto vinyl hose from bottling bucket or autosiphon.
Bottles
Beer bottles or wine bottles both work.
Figure you whether you want to work with caps, corks, or both.
Champagne bottles can work, too, but be aware that they require different sized caps and corks, as well as more specialized equipment.
Spring/Pump-Operating Bottle Rinser/Sanitizer
Convenient but not required. You can always just pour in sanitizer and swish it around by hand.
Quick way to sanitize bottles. Not extremely durable.
You can also use the heat setting on most dishwashers to pasteurize clean bottles before use.
Bottle Tree / Racks - For draining sanitized bottles.
You can also spray down a dishwasher rack with sanitizer and use that instead.
Closures: Caps, Corks, etc- Your bottles and closures will have to match.
Capper
Double-Lever Capper: There are two major models of double-lever hand capper out there. The Red Baron and the Black Beauty. They're more or less the same.
Single-Lever Capper: Made for bench top use. These are more reliable, since the same level of pressure gets applied uniformly around the cap, every time. They're also much more expensive. Some of these can be configured to handle both American (26 mm) and European (29 mm) caps.
Corker
Double-Lever Corker: Work best with short corks, or corks that are wet. Requires some muscle to use well. Poor at creating a uniform cork depth. Relatively cheap.
Floor Corker: Sits on the floor, has a single large lever. Very easy to use. Adjustable cork depth. Much more expensive. Some - but not all - of these are capable of corking champagne bottles.
Cosmetics
Capsules - These cover the corks in the neck of corked bottles.
Wax - Can be applied to capped or corked bottles. Makes nice gift bottles.
Labels - An exercise in creativity and practicality. Get a crazy as you like, or use masking tape and sharpie.
Kegging Equipment: There is no easier way to make sparkling mead. It's easiest to begin kegging by buying a complete kit - all of the hoses and clamps will be the right size.
Kegs. These come in ball lock and pin lock varieties. It's a good idea to pick one style and stick with it because you don't have to duplicate attachments.
Compressed CO2 Tank
Pressure Regulator - steps the compressed CO2 down to an adjustible, lower pressure suitable for pushing liquid from the keg or force carbonating.
Tubing and clamps - at the very least, you'll need two lengths of tubing. First, you need a length of gas tubing to attach the CO2 tank to the keg. Second, you need a length to run to a tap of some kind.
Picnic Tap or Jocky Box - for serving from the keg.
Beer Gun (optional) - for bottling pressurized liquid from the keg.
Kegerator or Keezer - makes forced carbonation much quicker and easier. Makes it easier to reach and remain at serving temperature.
Disassembly and Cleaning Tools - Most kegs are most easily disassembled with a socket wrench. The dip tube may need to be cleaned with a long, thin brush.