In a hospital, several patients became infected with a bacterium after surgery. the patients are not responding to antibiotics. by what type of disease are these patients likely infected, and why is it such a risk?

The type of disease that is resistant to antibiotics is called MRSA, methicillin resistant staphylococcus aureus. Drug resistant infections are highly risky because if an antibiotic treatment is found 1) there is a delay treatment; finding cultures & sensitivity takes over 14 days at best. During the waiting period then ineffective treatment is being used.
If the correct antibiotic is not found then the patient may risk infection spreading from the initial area of infection to widespread.
Patients are usually placed in some isolation to keep other patients from getting infection from cross contamination.
Medical Costs rise with more intense treatment.

By what type of disease are these patients likely infected, and why is it such a risk?

For the case presented above the type of disease that the patients are likely infected with is an Emerging Disease. It is such a high risk because it can spread fast making it even harder to contain and that they are resistant to many antibiotics.

I hope it helps, Regards.