There are 2 ways to propagate succulents, the first is Propagating by Division, in which new succulents sprout out from cuttings. And the second is Propagating with Offsets, in which many species of succulents will produce offset from the base of the plant.
Propagating by Division:
This method works best with plants that have grown too leggy. To start, cautiously pull out any leaves on the stem below the rosette. To do that gently twist the leaf off of the stem. It has to be a clean pull, leaving nothing left on the stem. If, you peel some of the stem that is fine too.
Once all the succulent leaves are removed, use sharp scissors or Pruning Shears to snip the rosette leaving behind a short stem.
To prepare the cuttings, cut off a piece of the cutting just above where the leaf connected to the stem.
Once you have your cuttings you’ll want to let them dry out a few days until the raw ends have calloused before you do anything else. After that you can Propagate the cutting in either soil or water.
How to propagate succulents in soil:
After the stems have calloused fill a trey or pot with Cactus or Succulent Soil Mix and place the cuttings on top of the soil. In a few weeks some roots will start to grow from the bass of the cutting. Don't water the cutting too much, wait until the root appear, then water once every week.
The parent leaf will wither after some time, remove it carefully, being careful not to damage the young roots. Leave your propagated succulents take root on their own. Then you can transplant them as you desire. Be careful not to place them in direct sun until the succulents are established.
How to propagate succulents in water:
After the stems have calloused, Place the cuttings on the edge of a jar or glass of water. With the raw end of the cutting just above the water surface. Put your glass in a sunny spot, in a few weeks roots will sprout out of the leaf reaching toward water.
Once the roots have grown you can keep growing the succulent in the water or you can replant them in potting soil.
Propagating Succulents with Offsets:
Some species of succulents like hens, chicks and aloe. And some types of cacti produce offsets.
Offsets are small plants that will grow from the base of the main plant. After the offset has been growing for 2 to 3 weeks, check the roots development and extract it from the main plant stem using a knife, or by gently twisting it until it come off. Avoid damaging the newly emerged roots.
Then follow the above steps for allowing the offsets to dray and propagating in water and soil. And developing the roots before you start replanting them.
And as a side benefit by removing the offsets from the main plant you are also helping improve it health and returning the growth energy to the main plant helping it grow stronger and bigger.
----------------
These tips probably won't apply to every climate in the us but this is what has worked for me. For even more detailed information on how to propagate succulents, check out this book Succulents Simplified.