"Anesthesia" is a wider umbrella term than most people realize with many levels of sedation.

Under "general anesthesia", the patient is completely unconscious. They don't respond to any stimuli. In rare cases, some patients may have an adverse reaction and still retain some sensation, but that's very uncommon. My understanding is that we are certain that patients are actually unconscious (and not just unable to respond) because none of the other involuntary responses to trauma occur during surgery: elevated heart rate, etc. In short, you are simply not there for a while. This is what you get for most kinds of significant surgeries unless the surgery requires you to be awake (like brain surgery where they may need to ask you questions).

"Sedation" or "twilight sedation" is a lower level of anesthesia. You are somewhat conscious and can respond to commands from the doctor. But you are unable to form memories of what's happening and you're usually on something like fentanyl that makes you entirely OK with whatever it is they are doing to you. This is common for procedures like colonoscopies and endoscopies where the procedure is somewhat uncomfortable but where you aren't being cut open.

In general, anesthesiologists are trying to balance the goal of patient comfort against the risks of deeper levels of sedation.