If someone is hurt at your house or on your property, the liability provision of your homeowners’ insurance policy will typically kick in to cover any injury claim that is filed.
When someone files a claim against you under your homeowners' policy, that is called a third party claim (by way of contrast, when you file a claim for your own losses, such as damage to your home from a fire or flood, that's called a first party claim).
Your liability coverage will pay all of the injured person's losses associated with the injury -- which means medical bills, lost income, pain and suffering, and other damages -- up to the limits of your coverage. And in the event that a lawsuit is filed against you by someone injured on your property, the insurance company will provide the defense for you.
So, if you’ve got $200,000 in liability insurance as part of your homeowners’ policy, your insurance would pay the injured person up to that amount. You could conceivably be on the financial hook for any amount above the policy limits, but that would be a pretty rare case involving very serious injuries. And even in cases where your liability limits are exceeded, your homeowners’ insurance policy may have an "umbrella" provision in place that provides extra protection in precisely those kinds of instances.
For more on the limits and types of injury liability coverage in a typical homeowners insurance policy, see this page.
Most injuries to visitors to your home will be covered under your policy, and most of those injuries will probably be based on accidental injuries that are governed by the legal concept of negligence. A slip and fall accident is a prime example of a claim that would be governed by negligence, meaning that the injured person would need to show that you somehow failed to exercise reasonable care in keeping your property free of hazards.
But with some injuries, negligence may not be required. For example, if you live in a "strict liability" dog bite state and your dog bites someone, that person can likely make a third party claim under your homeowners' policy without needing to prove that you were somehow at fault.
Your homeowners' insurance policy will likely not cover any injuries that result from an intentional act on your part. So if you assault someone and they sue you for the injuries, you won't have the coverage to protect you.