The ways of propagating your plants are various depending on type. If seeds are taken and planted that is classed as sexual propagation. This takes a while and the final quality cannot be guaranteed. Asexual propagation is where copies of the parent plant are produced through various means and is far quicker than using seeds. Some of the main ways of using asexual propagation are through leaf and stem cuttings, offsets and dividing.
Plants such as African Violet and Peperomia are easily propagated from leaf cuttings. Take a single leaf and place in compost. In a few weeks you find new leaves and roots forming from the base of the leaf.
Taking cuttings from the stems work well with plants such Geranium, Busy Lizzie and Fuchsia. Find a node, the small bump below a leaf. Then snip off a few centimetres from the tip. Remove all leaves, except for those at the tip. If you have rooting hormone dip the cut end then plant.
Many plants produce their own offspring that can easily be removed from the main plant to create new ones. Examples include tuberous plants such as Achimenes, succulents, Cacti, Zebrina, Birds Nest Fern and some Ivies.
For all the above propagation methods put the cuttings in damp compost and cover with a plastic bag out of direct sunlight. Then lightly water when dry.
If you have plants that grow quickly and are bushy with stems at the base. They are ideal for dividing. Remove the plant from its pot and gently pull it apart, use a knife if necessary, to form two or more plants. Then re-plant. Ideal candidates for this are things such as Spider Plants and Umbrella Plants.