Weight is a force, and therefore should be measured in units of force. But, of course, humans have very little experience with environments that don't have 1g of gravity, and thus routinely conflate force and mass. This kind of thinking goes far beyond aviation; you can find it everywhere. For instance, spring-loaded bathroom scales measure force but will almost always display a result in kg (in metric countries). It's sloppy, but it works, and so there's little incentive to change it.
The pound is both a unit of mass and a unit of force. A one-pound mass will produce one pound of force in standard Earth gravity. If for some reason you need to specify which one you're talking about, you can use the terms "pound-mass" or "pound-force" to differentiate*.
There's less of an excuse for this in the metric system, which has separate units for force and mass, but that doesn't stop it from happening anyway. If need be (such as when reading a manual that uses kg as a unit of force), you have to think of "kg" as "newtons divided by 9.8", because that's pretty much the way they're being used.
*Of course, just about the only reason you'd need to differentiate is if the local gravity field is something other than 9.8 m/s/s, which would mean that you're dealing with space travel, which would mean that you're almost certainly using the metric system to begin with, and thus would have no need to use either pound-mass or pound-force for anything.