The basis for making alcohol is simple. Yeast eats sugar and makes alcohol and CO2. All alcohol starts out that way. Wine is grapes or other fruit whose natural sugars are eaten by yeast and alcohol and CO2 is made, beer is sugars derived from grain like barley, wheat, and others which, again, is eaten by yeast and converted to alcohol and CO2.

There are more specific details...the yeast can only eat so much sugar...too much to start with will harm it. Yeast can only survive so much alcohol...too much kills it.

Wine is simply grape (or other fruit) juice that has yeast added and it ferments (the term for the yeast eating the sugars) over time. Once upon a time, the naturally occurring yeast in the air which would accumulate on the grape skins were sufficient to start the fermentation. Results were pretty random...sometimes it would be a good year, other times, not so good. Nowadays, winemakers kill the wild yeasts and add their own so they get consistent and predictable results.

Beer is similar, but there's an extra step. It comes from grain, but grain doesn't have naturally occurring sugars in abundance. When grain (particularly barley and corn) begins to sprout, enzymes develop. These enzymes convert starch to sugar. People learned to partially sprout and then halt the sprouting process to harness these enzymes, called "malting." The malted grain is then used to convert the starches into fermentable sugars...yeast is added and fermentation occurs.

Vodka and other spirits require one more step...distillation. Fermentation is the only way alcohol is MADE. Distillation doesn't make alcohol, it just concentrates it. To make spirits, you have to first ferment some kind of alcoholic brew like beer or wine or similar (in distilling terms this is referred to as "wash"). Once the wash has been brewed, it's put into a still where the alcohol is concentrated...it boils into a vapor, the vapor is collected and condensed back into a liquid...strong alcohol.

The brewing part is pretty labor intensive, whether you're making wine or beer. However it's perfectly legal to do so at home. You can either buy kits or start from scratch. Either way, you're allowed to brew up to 100 gallons per person of legal drinking age in your household, up to 2 people. The same applies to wine and beer (100 gallons of each per person...).
Distilling, due to safety and tax reasons is illegal to do in the home (ever) and not legal to do without a permit. The exception to this is New Zealand, Greece, and a handful of other countries that I can't recall at the moment.

 1 Super yeastObtain a packet of super yeast or distillers yeast. Only this type of yeast will be able to effectively and cleanly ferment table sugar and produce up to 20 percent alcohol. This kind of yeast also contains nutrients (such as vitamins and minerals) that will support the growth of the yeast and enable effective fermentation. Sucrose and other kinds of yeast do not contain such nutrients.

2 Choose the fermentation vessel. For 5.5 to 6 gallon (20-23 L) batches, a 7.5 gallon (28 L) food grade bucket and lid works very well because it is easy to pour the ingredients into the bucket and stir them, and because the bucket has a large capacity. Glass carboys can be used as well, and for smaller batches, use one gallon jugs. However, a drilled rubber stopper that fits the airlock will be needed that fits the carboy or jug. Leave a good amount of extra head space, equivalent to about 1.5 to 2 gallons, in the vessel for foam and gases that can form during fermentation. The lid for the plastic bucket should have a hole drilled in it and a rubber grommet that the airlock will fit in during fermentation. The lid should also have a rubber gasket in it that will create an airtight seal between the lid and the rim of the bucket.

3 Clean and/or sanitize the equipment. The fermentation vessel (and rubber stopper for glass vessels or lid for a plastic bucket), airlock, and a large spoon should be clean and sanitized. Cleansers may not be needed if the equipment is new. Use a sanitizer such as iodophor[1] that is made for brewing and winemaking. All these items are available at homebrewing and winemaking shops. Fill the fermentation vessel to the brim with sanitizer.
Sanitize the plastic bucket lid (or rubber stopper for a carboy) and airlock in a clean vessel. The plastic lid will won't fit in the plastic bucket, so sanitize it in a large pail or pot. The airlock and spoon can be sanitized in the bucket if desired.

Sanitize the plastic bucket lid (or rubber stopper for a carboy) and airlock in a clean vessel. The plastic lid will won't fit in the plastic bucket, so sanitize it in a large pail or pot. The airlock and spoon can be sanitized in the bucket if desired.
 

4 Determine how much sucrose (common table sugar) to use. More sucrose will result in more alcohol, but the yeast will only be able to ferment a certain amount of sucrose. When the total desired volume is 5.5 gallons (20 L), no more than 17.5 pounds (8 kg) of sucrose should be used, as this should achieve the maximum of 20 percent alcohol. Some kinds of yeast, however, will not be able to achieve 20 percent alcohol, so less sucrose must be used, resulting in a more dilute solution of sucrose and water. The packet of yeast that is used will have directions that indicate how much sucrose can be used. If making two batches, be sure to use twice as much yeast (two packets).

5 Mix the sucrose and warm water in the plastic bucket or a sanitized metal pot to dissolve the sucrose. Tap water or bottled water can be used. Fill to about 5.5 gallons (20 L). The water should be about 90 degrees Fahrenheit (32.2 degrees C). The yeast is a living organism and requires a certain temperature to function properly. Use a metal pot to avoid scratching the inside of the plastic bucket, as scratches can harbor bacteria and make the bucket impossible to sanitize. When all the sucrose has been dissolved, carefully pour the sucrose and water solution into the plastic bucket or glass carboy.

It is not necessary to sterilize the sucrose solution before fermenting, but if desired this can be done by boiling the sucrose solution for fifteen to twenty minutes. Take into account that some of the water will evaporate, so add a little more water before boiling. Cool the boiled mixture with a copper chiller that is designed for cooling homebrewed beer, and sanitize the chiller by boiling it in the sucrose mixture for fifteen to twenty minutes. When transferring the cooled solution to the plastic bucket, be sure to let the solution splash and pick up air, as it needs to be aerated to ferment properly. Boiling will remove oxygen, but tap water should be adequately aerated. When boiling the solution, ingredients such as herbs can be boiled with the mixture to imbue color as well as flavor, but the flavor will change when the sucrose solution is fermented by the yeast. This is because the yeast produces carbon dioxide that will strip out flavors.

6 Add the yeast. Open the yeast packet and dump the yeast into the sucrose and water solution. If using a plastic bucket, stir somewhat if desired to homogenize the mixture. A sanitized, dry funnel can help to prevent a mess when adding the yeast to the narrow opening of a carboy.

7 Affix the lid to the bucket. If using a plastic bucket, tightly push the lid onto the bucket so that an airtight seal is formed. This may be somewhat difficult, and may require some leverage.

8 Affix the airlock to the lid or carboy. Push the airlock into the lid if using a plastic bucket. If using a carboy, push the airlock through a drilled rubber stopper an fit the stopper snugly in the mouth of the carboy. Add clean water or vodka to the inside of the airlock so that carbon dioxide can be released from the fermenting solution while air is kept out. There are two kinds of airlocks, bubbler and three-piece airlocks. Both work very well.