Natural pest control
Natural pest control starts long before a 'pest' arrives in your garden. In fact, when dealing with 'natural pest control' the last part of the discussion is about 'what to spray' or 'how do I kill (insert insect name here)?"
Natural pest control seeks to avoid trying to fight it out with nature; instead it seeks to cooperate with nature and not conflict. Plants that are growing with the right amount of moisture, sun and nutrients usually avoid most pest infestations. And in 'natural pest control' a small amount of insect infestation is not necessarily a cause for alarm or destructive action.
A plant that is being constantly attacked by aphids has something wrong with it that is attracting aphids and, while you can spray insecticidal soap, horticultural oil or some other pesticide to rid the plant (and yourself) of aphids, unless the underlying cause is addressed, you will have to deal with aphids on this plant again. Better to understand why the plant has aphids and fix THAT before spraying or otherwise attempting to deal with the aphids. (By the way, aphids are often attracted to a plant that has been over-fertilized because the excess nitrogen has created an excess of soft, lush growth that aphids find irresistible.)
The first part of natural pest control, then, is to grow a healthy plant.
But even a healthy plant might get a few insects. What then? If it's not a huge infestation, why not leave them? The idea that there can be no insects or insect damage on a plant comes from folks who grew plants for show. In the past, the only experts on roses were the folks growing roses to win ribbons at a rose show; the experts on daffodils took their blooms to the daffodil show and so on. They were the experts that wrote the first books on how to grow this or that plant and their focus was on the PERFECT rose or daffodil and they could brook no damage to their prized plants. If we aren't going to take our plants down to the plant show and try to win first place, why do we care if there is a bug here or there or that this leaf has been chomped a bit? Honestly, a chomped leaf or a slightly disfigured flower will harm a person a good deal less than spraying a bunch of pesticide about - even the organic pesticides.
At some point, however, a plant might get an excessive infestation. You have decided there are too many insects and you are uncomfortable with the situation. You might consider giving the plant the heave-ho. Bugs and plant go off to the happy compost pile in the sky and that's it.
In natural pest control, you have almost completely solved 99% of your pest problems by using one of the solutions above. It is apparent that spraying insecticide is not a high option on the list.
And with good reason too. The 'non-organic' pesticides don't come with high recommendations in their interactions with other species, fish and man among them. They also have been implicated in damage to other species further up the food chain of the pests you are trying to eradicate. Insects are preyed upon by other insects, birds and rodents and the insecticide you spray today will persist (in fact, is formulated to persist!) and does not simply vanish. The insecticide sprayed today will become a hazard to other species.
The organic pesticides, on the other hand, are often thought of as 'safer, kinder' because they do not persist. Most of them are only active while still wet. But even at that there are problems. Most organic pesticides are non-pest specific, so while they won't be killing anything tomorrow, they will kill any insect that comes in contact with it while it is still wet. Honey bees, which have been high on every gardener's mind because of the wholesale destruction of hives through out the US, can be killed off wholesale by spraying an organic insecticide at the wrong time of day.
Grow a healthy plant. Pack the garden with diversity and try to have blooms throughout the growing season to attract lots of insects. If a garden is blessed with a large number of insects of all different species, chances are the insects themselves will keep populations in check. And there is nothing more natural than allowing insects to control themselves.
In a nutshell, that is the basic idea of natural insect control.