Capacitor swelling is characterized by a visible bulge or deformation on the top or sides of the capacitors. This deformation is a result of the capacitor failing or being under undue stress, often from heat or age. It's important to identify this physical change as it can lead to the malfunctioning of the motherboard and potential damage to the system. The presence of a burning smell would usually be related to other types of component damage or electrical short circuits, and the system not powering up could be due to a variety of other factors, not specifically related to capacitor swelling.