With our challenging environment here in the desert, it's surprising to me that weeds would even get established here, but of course they do. When I was teaching a weeds class recently, someone asked me for a good definition.
My personal definition is "a weed's a plant that's growing where you want something else, and it's doing better than the things you did plant".
You could consider any plant to be a weed if it's growing where you don't want it. Or when. For instance, if you had carrots in your garden last year and they came up again where you now want your tomatoes, then those carrots are weeds.
Usually though, weeds are plants that appear and you never want them. Things like Russian thistle, tumbleweed. It's a seriously awful plant, and while there might be some creature somewhere that likes it, most of us find it a horrible, invasive and noxious weed.
I had a class describe how they'd build a perfect invasive weed. It could have many seeds, or reproduce through additional, asexual means. It'd be able to survive a range of environments, spread easily. It'd have minimal requirements. In some ways, that might sound like a plant we'd want to have in our own gardens.
But it brings us to the question, what is it about these plants that allows them to become weeds? There are several ways. A number of weeds got carried in as a contaminant in grain, or on an animal. Nobody ever said that weeds had to be ugly, and some of them started out as lovely ornamentals. For instance, salt cedar was brought into this country from Asia to help stop soil erosion and as a garden shrub. Now, it's a noxious threat that crowds out native plants all over the west, uses large amounts of water and turns our salty soils even saltier.