There is nothing wrong with the brake but it is slightly rusted and badly out of adjustment.
Loosen the brake bolt and tighten the centering screw so the top of the head stands a bit proud of the caliper. Turning the centering screw to the right (tightening) rotates the caliper to the right, but it also shifts the right pad down and the left pad up, and it brings both pads closer to the rim. You might need to loosen the cable.
Next, realign the pads so they hit the rim squarely. The top should be a millimeter or two below the edge of the rim and the bottom should not be below the machined braking flat. You don't want the pad rubbing the tire, and if it protrudes below the braking surface, it will form a hook as it wears. The hook will get stuck on the rim when you release the brake. If the pad is too worn for this adjustment, replace it.
Next, center the caliper the best you can by hand and tighten the bolt. Then use the centering screw to fine-tune the centering so the two pads hit the rim simultaneously.
While you're at it, replace the brake cable and housing--they're rusted and worn. Also drip a bit of thin oil, such as Tri-flo, into the pivots, and wipe the whole caliper with a slightly oily rag. This will keep the corrosion at bay.