This is a beginner's guide to propagating succulents. I tried to be succint, but this is detailed topic. There are caveats and exceptions to just about everything. I won't be covering them for simplicity. I am an enthusiant, not an expert, and possess mostly practical knowledge. Corrections are appreciated.
Hens and Chicks The most popular, these are identified by the thick, fleshy leaves, barky stem and a common habit of crawling across the ground with shallow roots.
Fleshy rosettes Aloes, haworthias and agaves fall in this category and share a similar appearance to hens and chicks - the rosette. These are easily distinguished though because they generally don't have a stem. They are all leaf.
Fleshy stems Think traditional cactus, with arms. Sometimes these have leaves, sometimes they don't.
Epiphytes Epiphytes are plants that often grow in trees and get nutrients from the air and rain.
Hens and chicks can be propagated by leaf or by cutting.
Leaves don't fall off the plant naturally. Each leaf represents stored water and energy and the plant would rather use that for itself than anything else. The leaves have to be physically removed from the mother plant. You SHOULD NOT remove leaves from a healthy plant at a nursery as that would morally constitute theft.
The leaves are attached loosely as they have evolved to drop off and make a new plant. Hold the leaf close to the base and gently twist. It should leave a smooth end with no tearing. Demonstration. Healthy leaf removed
Now you have leaves. Place them in a shady, cool, dry location to cure for a week. During this period the leaf has exposed flesh and it will callous over. If you place it in the dirt or sun without a curing period, it may die.
After a week you can place the leaf directly on top of the dirt and the roots will find their way down, or you can plant the end of the leaf in the dirt and it will sprout. Watering should be misted only until the plant is established to avoid damanging the new roots.
At any point the stem can be cut and the rosette planted or even just set on the dirt and it will root. The original plant will also regrow. A curing period doesn't seem to be necessary for stem cuttings, but if a plant is beloved you might consider curing to be safe. Cutting & Planting
Fleshy rosettes, including but not limited to aloes, agaves and haworthis CANNOT be propagated by leaf. They propagate by sending out pups. Pups like chicks, except they spread via the root system underground. You propagate them by repotting the entire plant and separating the root ball. It makes the most sense to do it when the plant is root bound with many pups.
Root bound aloe. It was so root bound I had to cut the pot away.
Take the plant out of the pot and loosen the dirt. Unlike other plants succulents don't need the old potting dirt to protect the roots. Use your fingers to gently tease the roots until you can separate the individual rosettes. Then plant them in new potting mix. Wait a week to water in case any roots were damaged. This gives them time to heal.
I planted a handful of them in a new pot, but I took most of them without separating and planted them in the ground.
Not all fleshy stemmed succulents are cacti. If you know the difference you probably don't need Prop 101, so let's move past that. From a practical perspective, they're the same.
Cacti can be easily propagated by taking a cutting. You simply choose a piece and cut it off at the appropriate length (6-24" depending on diameter). For segmented cacti like opuntias take entire segments. Ideally you would not do this during the growth season. Cacti should never be disturbed while growing, identified by orange spines at the head. That's best practice, however the more robust species can take anything you can throw at them.
It's recommended you always use a clean knife to limit the chance of infection, however the larger cactuses have a dense stem or can become barky. I've used a machete, a lopper and a straight up saw to cut mine and never had a problem, however I've only cut the most robust varieties. Do this at your own risk.
After cutting, leave the segment in a cool, dry, dark place for a week (or two for the big ones) for a callous to form over the wound. Then you can plant the cutting directly in the ground or a pot. Put enough under the soil so that it's stable, but do remember when repotting it later that THERE ARE THORNS under the dirt line. Large cuttings can be staked for stability (opuntias for example).
The original plant will grow a new arm below the cut.
This cereus was cut into multiple pieces about 2' long. I put it in a 3 gallon pot. It will last one growing season in that pot and then will need to be planted. It's too large and fast growing to be kept in a container for very long. Because it was cut at both ends it will send out arms.
Epiphytes are plants that draw nutrients and moisture from the air, rainfall and decaying matter built up on trees. They are succulent plants but many are not what you would traditionally think of as succulents: air plants (tillandsia) and bromeliads for example. These can be propagated by pups the same as an aloe. For air plants, you just separate one and tie or glue it to a tree or piece of bark.
There are more traditional cactus/succulent epiphytes as well. The Christmas cactus (schlumbergera), dragon fruit (hylocereus) and the genus epiphyllum are examples. These can be propagated by cutting. These plants are capable of growing in the ground as well as on trees, and will even send roots down a tree into the ground. This gives two methods of propagating by cutting.
Plant it like any other cactus. Put it next to a tree, in the ground or a pot. Tie it to the tree to guide it.
Tie the cutting to the tree and just wait. This takes longer than planting it, but eventually it will send out epiphtic roots and new segments.
For further reading I recommend The Complete Book of Cacti & Succulents by Terry Hewitt.
