Succulent babies, offsets, and offshoots are all the same thing. Your treasured succulents are doing so well they have decided it's time to give birth.
Congratulations! You have mastered the art of proper watering, the best soil, getting them just enough sunlight and copious amounts of neglect; all things succulents yearn for.
You"ll notice this new growth typically when these plants are in their active growing phase. For most, this will be in the Spring and Summer months. A few will occur in Fall and Winter. Blooms and offsets will usually occur at the same time.
To ensure their success, know when and where to look, know your plant's offset growth tendencies, be patient, and what to do next.
When Succulents Produce Offsets
As mentioned before, most succulents will produce babies in the Spring and Summer months.
Echeveria, Graptoveria, Haworthia, and Sempervivum (to name a few) will all produce new plants if they are doing well. These will occur in Spring and Summer.
Aeoniums will produce many offsets in the Fall and Winter months.
If you start noticing new blooms, know that there's a good chance offsets or offshoots are likely coming. These plants will keep you guessing, so don't expect blooms to occur first. Offsets can come before blooms.
Also, just because a plant blooms doesn't mean it will produce new babies.
Where Succulents Produce Offsets
With the exception of Sempervivums, most offsets will be found near the base of the soil at the bottom of the plant. These come right off the stem, and there might be one, two or even three (if you're lucky).
Sempervivums (image below, left side) are much easier to identify. They will roll away from the mother plant. They are attached by a stolon, a stem that feeds the offshoots all it needs to grow.
There will usually be two - eight new babies that form around the mother Sempervivum.
Know Your Plant's Offset Growth Tendencies
It's important to know that babies, especially the ones that grow at the base of your mother plant, will not affect or harm them.
The offsets might look squished or uncomfortable, but know that they are exactly where they need to be.
Trust the process of Mother Nature. They've been doing this for much longer than we have.
Remove the succulent babies prematurely, and you put it at risk to fail.
I recommend waiting until the offsets are about half the size of the mother plant before removing them. This ensures your babies have the proper nutrients and best chance for life on their own.
Once your succulents are producing offsets, you should consider repotting them in a slightly larger pot to accommodate the hen and new chicks.
Once the offsets are half the size of the mother plant, you can cut them off using a pair of pruners.
Wait for the cut to callous over. Place them on top of fresh soil, don't water, place them in a shady, but bright area, and neglect them.
Soon enough, they'll root into the soil and voila! You have successfully made one to eight new plants to play with.