Yes, the little ones, offshoots or pups, are indeed new Century Plants, although they are still connected to the mother plant. Since none of the 6 Century Plants, all members of the genus Agave, are native to the area around Liberty County TX on the upper Gulf Coast, we are going to use Agave americana (American century plant) as our example. The propagation and care of all the agaves are pretty much the same, so it will work fine.
According to our Native Plant Database page on Agave Americana, propagation can be by both the pups or seeds. If you want to have more of the plants, the offshoots are probably your best bet, as seed germination can be pretty slow. From a previous answer on transplanting offshoots of an agave:
How do you transplant an Agave? Like kissing a porcupine-very carefully! All agaves reproduce by pups in about the same way, so these instructions should apply to your plant. You may want to begin by cutting away some of its menacing leaves from the "Mother" plant where you will be working.
First, and no kidding this time, approach this task with care. The agaves have survived in very hostile environments by being pretty hostile themselves to grazing by livestock or digging up to clear land for farming. You can, indeed, make new plants of the "pups", but first, protect yourself with heavy leather gloves, and possibly goggles. With clippers, remove the fierce spines on the pup before you separate the plant from its parent. If there is a clump of several new plants, gently break them up by hand or with a knife. One standing alone can probably be popped out of the ground with a trowel or small shovel. Again, careful, you are close to a very forbidding parent plant that won't hesitate to get you in the eye or the arm or the back with those long, sharp-tipped leaves. Get rid of loose roots and then, as if it was an onion, peel away leaves until you get to the best-quality leaves in the center of the rosette. Carefully discard all the removed spines and leaves where someone won't come along and step on them. And the compost pile is out, you don't want to stick your hand into that!
If it was necessary to cut the transplant, you need a clean cut on the base of the wound. You can dab the wound with sulphur before putting the cutting in a pot filled with a "cactus" potting mix. Top off the pot with more sand or "cactus mix", pack it down and put it out of doors in full sun. Don't overwater it and in a couple of months it should be ready to be on its own.
Now you're ready to decide on the permanent location for the new plant. Since it can be anywhere from 8 to 40 years before the plant summons up the energy to bloom and then it dies, you should not worry about where it can bloom best, but where the plant can spread out and the plump leaves be seen best. And, of course, where it's out of foot traffic or where a child or pet might blunder into it. You should be aware that the agaves should be protected from teen winter temperatures to avoid damage. A short spell of freezing weather shouldn't harm it.
If you choose to keep the new plants indoors in a pot, the agaves are so slow-growing that you shouldn't have to transplant into a larger pot very often. They can do very very well indoors in a good light from a window. Remember, they are succulents, and like most other succulents, they need less water and can tolerate quite a bit of shade. For more information on container gardening, read this article from the Lady Bird Johnson Wildflower Center "How To Articles".
From our Native Plant Image Gallery: