Benlormat has it spot on. Although various definitions of weeds are floating around by far the most widely accepted one, and the only one that works, is “a plant where it is not wanted”.
And it really is that simple. Far from being plants with no redeeming characteristics many of the world’s worst weeds are plants of great beauty. It was there attractiveness that led people to move them from their natural environment in the first place, and in many cases once they had been moved they rapidly became weeds.
However weeds can also be indigenous to an area. Most of the world’s weeds fall into that category. An oak tree in a wheatfield is indisputably a weed, which is why so much effort was spent eliminating them before cultivation.
And as you can see there are various categories of weeds. There are economic weeds such as the weeds that compete with commercial crops and cost money in a very direct sense. There are environmental weeds like the punkwood invading the everglades that don’t cost money directly but that destroy the ecological value of land. There are aesthetic weeds like crabgrass that are unwanted simply because they are considered unattractive. There are medical weeds like poison ivy that are unwanted because they pose a physical risk to people or animals. There are also several other possible classes of weeds that don’t fit neatly into any of those categories.
The simple fact that a plant is attractive doesn’t in any way stop it form being a weed. An attractive plant can even manage to be an aesthetic weed, as odd as that might seem. While baby’s breath might seem pretty in isolation I’m sure you would think differently if your entire lawn were choked with it.