How do I propagate aloe from cuttings at home?

The easiest way is to break off a side shoot at the base of the mother plant and let it dry out for a couple of days in the air. The cut should seal over with a film, otherwise it'll start rotting in the soil. After that you can plant it in dry succulent soil or regular soil mixed with sand - the main thing is that the soil is loose and doesn't hold moisture. Pick a small pot with a drainage hole.

Don't water the aloe at all for the first two weeks, let the roots grow in. Then start watering rarely and sparingly - only when the soil is completely dry. The plant doesn't like moisture, so overwatering is more dangerous than drought. Keep the shoot in a bright spot but not in direct sunlight until it takes root. Usually after a month or two new leaves appear, and you can tell everything went well.

If you don't have a shoot, you can propagate from a leaf - tear off a healthy leaf, dry it for several days, then put it on damp sand and wait for roots to appear. But shoots are faster and more reliable.