Weeds are just plants we don't want. Nature makes no distinction between weeds and 'plants', so in weeding we face the truth that gardening is the first unnatural pursuit of humanity.
But lets not get hung up on the philiosophicals. Lets get the hoe out!
We need to control weeds in order to prevent them competing with the plants we do want, for the moisture and nutrients in the ground.
By going regularly between rows of plants with a hoe we can remove the weeds. This is (in case you hadn't worked it out) the reason we plant in rows. it makes hoeing weeds easier.
If you do your weeding on a warm day, the little weeds you pull to the surface will die off in the heat. If it is a wet day it is still worth weeding, but you might want to remove the weeds you bring up to the surface.
Bindweed and mares tail have deep roots and are a long term problem. However, constant removal will eventually bring even these pernicious weeds under control.
Weedkiller is how some people chose to solve their weed problem, but please be aware of the environmental issues involved and thing long and hard before going down this route.
Sometimes you just need to get rid of weeds. For instance, when you move house the new garden may be a mitre deep in weeds. On these occasions strimmers are the gardeners best friend.
Of course if weeds are growing well, why not encourage and then harvest them as a crop? For example stinging nettles make excellent nettle wine.
Dandelions also make excellent ingredients for ales and their roots can be used as a tasty vegetable.