The process of producing new plants is fairly simple - I will break down the process for indoor and outdoor plants.

Indoor Plants >>> Many plants like Aloe Vera - produce hundreds of their own little babies (or pups) regularly from the mother plant.  Simply separate the baby from the parent and pot up in new compost.  Many plants like Jade plants readily propagate from a cutting from the mother plant.  Cut off a 4 inch stem and place the stem in a pot of compost.  For say an African Violet.  Cut off two healthy leaves (with stems) and place the cuttings back to back in a small pot.  Keep the pot watered from time to time until the plant starts to produce new leaves.  Often if one method fails - try placing the stem in a jar of water.  Leave it on a window sill and in time the plant will often produce new roots.  Pot up the plant once it has a strong root system.

Outdoor Plants >>> There are 2 types of cutting.  Softwood and Hardwood. 

Hardwood Cuttings : 

are taken towards the end of the growing season when the plant wood becomes firm and woody.  To get a better success rate take lots of up to 5 inch cuttings.  Prepare the cutting by cutting below a node (the place where leaves come out).  Then trim the top of the cutting above a node.  Place the cuttings into a terracotta pot around the edge.  Rooting hormone is optional but not essential.  The cuttings should be ready to pot up after the next growing season in the following spring.

Softwood Cuttings :

are taken about 6 weeks after the first green shoots in spring.  So for most places softwood cuttings start in June and run for the summer.  They are taken in much the same way as hardwood, but take root much more readily with the warmer weather.  Keep cuttings regularly watered and pot up in about 2 to 3 months.

All cuttings should be pruned regularly in the early stages to produce a fuller more compact and healthy plant.  This can be done by pinching out the growing tips regularly.  Which promotes bushier growth.