There are a few reasons why a mesh might not move/deform after being rigged. Here are some things you can try:
- Remove duplicate geometry
- Make sure the mesh hasn't already been parented
- Check your armature modifier settings
- Check if you're in Rest vs Pose Position
Armatures don't work well, or at all, with overlapping vertices. Select your whole mesh in Edit Mode, press M, and choose By Distance to get rid of overlapping vertices.
Clear existing parents
Usually, parenting a mesh to an armature will clear existing parents. Occasionally, it doesn't, but it's super easy to do yourself. Just press Alt P. Note that you'll need to re-parent the mesh to the armature after doing this.
Fix modifier settings
By default, the Armature Modifier settings are correct. If you're working with someone else' model, however, you may have an armature with incorrect settings. The most important of these is the "Bind To Vertex Groups" checkbox- this must be turned on for the armature to do anything.
Another important one is the "Display Modifier in Viewport/Render" buttons. This may seem obvious, but if these aren't turned on, the effects of the modifier (and by extension, the armature) will not be visible. It's easy to miss these buttons, don't skip this step!
Switch to Pose Position
This is probably the trickiest one on this list- it's not difficult to miss, but if you don't know it exists, it's very difficult to solve.
If this toggle is set to Rest position, your Armature will go back to the default pose in Edit or Object mode, and your mesh will follow. If your mesh is designed to fit your Armature (as it should be), you might think your mesh isn't moving at all, since that's what it looks like visually. Try switching the toggle to Pose Mode, and you'll see your mesh and armature jump to your set pose.