I have had a fascination with these unique plants since I was eight, and I started collecting as many species as I could around seven years ago. I now have three huge planters filled with specimens, as well as countless arrangements.
This is a summary of what I have learned over the years, if you have any questions, feel free to ask them in the comments.

The term “succulent” refers to a type of plant that usually has thick fleshy leaves or stem. These plants are built for water retention and surviving droughts. About 60 different plant families contain succulents.
Most cacti are botanically succulents, although they aren’t usually referred to as such.

Succulents have the ability to thrive on limited water sources, such as mist and dew, which makes them equipped to survive in an ecosystem which contains scarce water sources. Succulent plants may store water in various structures, such as leaves, stems, and sometimes roots.

Many succulents come from dry areas such as steppes, semi-desert, and desert. High temperatures and low precipitation force plants to collect and store water to survive long dry periods. Some species of cactus can survive for months without rainfall. Succulents may occasionally occur as epiphytes - "air plants" - as they have usually no contact with the ground, and depend on their ability to store water. Succulents also occur as inhabitants of sea coasts and dry lakes, which are exposed to high levels of dissolved minerals that are deadly to many other plant species.

Step 1: Watering and Fertilizing

During the growing season, a balanced fertilizer, which has been diluted to 1/4 strength, can be added to the water for each watering. (A balanced fertilizer is one that has roughly equal proportions of Nitrogen, Phosphorous, and Potassium. A 10-10-10 fertilizer diluted to 1/4 strength is ideal.) For most, the period of growth is from Spring into Fall. Many plants rest (stop growing) from late Fall to early Spring, when temperatures are cool and daylight length is short, and during mid-Summer, when temperatures are at their peak.

Many believe that succulents should be watered slightly, every once and a while, but in reality, they won’t thrive under those conditions. Succulents, in fact, should be watered at least once a week, soaking the soil completely, without allowing it to puddle.

When the weather cools and day-length shortens, plants enter their rest period. During that time, increase the interval between watering, and let the potting mixture dry out between watering. Some people say that during dormancy, cacti and succulents should be given just enough water so that they show no sign of shriveling. Use some common sense here. If your plants are kept indoors on a window sill in a heated room during the Winter, they will need more water than if they were over-wintered out-of-doors. In any case, do not fertilize your plants during dormancy.
Tap water often can be alkaline and/or hard, meaning it contains high concentrations of dissolved minerals. Such minerals can build up in the plant's 'soil' over time, causing harm. This is one good reason why your plants should periodically be 'repotted.' Buildup of such minerals can also cause unsightly deposits to form, especially on unglazed clay pots. Never water your plants with water that has been through a softening system that uses salt as a recharging agent, as these systems simply replace the "hardness" in the water with sodium ions. Rain water is preferable to tap water, if you can manage to collect and store it.

Step 2: Lighting

Most succulents like bright light, but not all will tolerate intense sunlight, especially with high temperatures. Telling whether or not the sunlight is at an optimal level requires a knowledge of the plant when looks “normal.”
As you will quickly learn, the more precise care (or less precise, as it may be) of this type of plant is based mainly off of instinct and “feel.”

When receiving too much sunlight the plant can appear off color or “bleached,” though it could also indicate a lack of water. If your plant is receiving too little light, it might etiolate and/or appear to really reach for the light source. (Etiolation is the condition where a plant becomes "drawn," for example, a cactus plant that is normally round begins to look as if it is being stretched out from the growing point at its center).

Your plant will suffer if left in such light conditions for very long. When transitioning such a plant to stronger light, keep in mind that it will be especially prone to scorching, so make the transition slowly. The plant will naturally grow toward the light but when it is warped and bent in an unnatural way, something is wrong

Step 3: Potting

To start off, most succulents do well in most any type of pot, although drainage holes are extremely recommended. Though it may seem like a pain (and I agree), it is advised that every year your plant be re-potted, either to freshen the soil or give the roots more space. Also if it outgrows the pot, although I feel that that would be a given.

Cactus potting mixes are usually available commercially, but many people like to create their own mixes. Pretty much the most important part of the soil is that it should drain very well. To achieve this, you add horticultural-grade sand and grit to the compost potion of the soil. A good starting ratio for the mix's components are one-third compost, one-third horticultural-grade sand, and one-third grit.
For the compost, peat should not be used because it promotes parasites, and doesn't contribute to nutrient levels. Many people start with a commercial potting mix for the compost component. The sand component should be horticultural grade, relatively coarse, and sharp. Never use non-horticultural grade sand, as this is usually not washed, and can contain salt. For the grit component, most people agree that horticultural pumice is the best. Some also use fired clay product such as natural cat litter. If using one of the clay products, you must ensure that it is a fired clay, so that doesn't break down when wet.

Step 4: Dealing With Parasites

Parasite care is pretty easy, simply wash with with rubbing alcohol or if you want to be more specific and target only bugs, than some insecticide will do. Spotting parasites is also fairly easy, look for small white critters moving on the plants, or splotches, which indicate fungus. You usually only need to check for them if the entire plant, or a group of plants, look sick.

Step 5: Propagating Via Clippings

The clipping of most (but not all) succulents can be grown into entirely new plants! This makes it easy to expand upon your succulent empire without having to go to the store! When deciding what leaves to trim, you should try to pick mature leaves near the base of the plant. For plants with many long stems, you can simply cut the stem into segments, and most should grow.
When taking leaf clippings, pull the leaf off gently, and be sure to break if off the stem. Leaves that are cut off, or broken, usually do not grow.

When planting, you can stick them into the dirt, or lay them on top, be sure that the base of the leaf touches the substrate.

Step 6: For Those With Terrarium Succulents...

Terrariums can be beautiful works of art, and a great addition to the home, but be wary, you may inadvertently injure, or even kill your plants. You must be VERY careful with the amount of water and sun it gets, as too much can drown the succulent, and too much sun can roast it. Do not water unless the soil has been dry for a period of time, and water only enough to make the soil damp. Keep out of direct sunlight, as the enclosure will trap the heat.

Enjoy your plants!