What happens if a party who's bonded doesn't or can't complete their work? There's no single answer to this, but if you're the party that issued the contract (the "obligee") you should pat yourself on the back for getting the performance bond in the first place. If not for the performance bond, you'd be out of luck and have to complete yourself, and then try and recover any overage against the non-performing party. But if there's a surety bond, the surety will at least come into the picture, although that doesn't mean the surety will necessarily complete the work for you.
The surety's rights and obligations will be dictated by the wording of the performance bond, which like any other contract will be enforced as written. There are some form performance bonds, like the AIA performance bond, but many other individualized forms are used. Remember, the surety's obligation is limited by the "penal sum" or amount of the bond, less the value of accepted work to date reducing the contract value. That means one option of the surety is to do nothing and just pay any remaining balance that can then be used to offset the obligee's cost to complete.
More commonly, the surety will either "tender" a completion contractor who will then be put under direct contract to the obligee under the bond or will hire its own "takeover" contractor to complete the work through the surety. Both have pros and cons, although most sureties will often prefer the tender approach, which often is of benefit to the obligee to as it involves one less party in completion.
As in most completion circumstances, there's no single approach and the best approach will vary depending upon any number of facts. Again, the good news is there is a performance bond though, which as least provides those various approaches for consideration. I'll talk more specifically about these various approaches in later posts.