Weeds are only plants where you don’t want them or a wild version of something you might like growing in the garden - think Achillea / yarrow; geranium / cranesbill – they all look right in the right place. And it may be counter-intuitive, but leaving a few weeds undisturbed can attract a greater variety of beneficial insects to the plot.
Fat hen was eaten by Iron Age people as a grain, and the leaves are edible too. It was commonly eaten until the 16th century, when it began to be replaced by things like cabbage and spinach, but it is still cultivated in parts of India as a food crop. Ground elder was reputedly eaten as salad by the Romans and of course nettles can be used for soup, not to mention a fertiliser: visit nettles.org for instructions on making a plant food from nettles (if you can stand the smell while you brew it)
Plenty of other weeds are still eaten today. According to treehugger.com the following weeds are edible: dandelion, purslane, clover, lamb’s quarters, plantain, chickweed, and wild amaranth. And there is much advice on the web about uses of edible weeds: see, for example abc.net.au
Apparently dandelion roots can be used for a coffee substitute (I did try this once: I confess I thought it was pretty awful, but that’s only my take!).
Some say that really pesky weeds such as couch grass and bindweed should just be burnt. But others believe that, provided they are dried out before being added to a compost heap, followed by covering the dried pile for a few months, they can make very good compost. As a non-driver with limited ability to take heavy stuff offsite, I include practically all weeds in my compost heap (woody material goes to the Burn Bin of course – more shameless plugging). It all rots down eventually to usable compost and I don’t think this causes my plot to be any weedier than average.
Deep rooted plants like dandelions and docks can bring nutrients to the surface. Chickweed, if allowed to grow in autumn and dug in during the spring, can act as a green manure, reduce rain splash and reduce leaching of nutrients.
If weeds never grow on your plot it might indicate that your soil has problems. They can also tell you a bit about the type of soil you have and how it is working: horsetail, creeping buttercup and silverweed will grow where there is poor drainage; clover and vetch thrive on low nitrogen; and sorrel and plantain like acid soil. By contrast spurge, chickweed and groundsel like humus-rich, well aerated soil which is higher in nutrients.