I've generally heard the problematic plants described as
"invasive non-native". Maybe it's easier just to call it a weed. There are a few invasive natives, but they're really rarely trouble makers.
We have a native thistle which is not a problem, but also have the non-native scotch thistle. They can be invasive, taking over and crowding out native species.
Like Stone, I do let a few thistles grow because the birds and bugs love them, but have to be careful not to leave too many.
Siberian elm trees are a considered a "weed tree" here, an example of an invasive non-native that can crowd out the locals.
In parts of California, Eucalyptus are considered undesirable invasives!
