There are two types of anesthesia - general anesthesia and local anesthesia. General anesthesia results in full unconciousness in your cat resulting in no sensation anywhere in the body for an indefinite period of time. Local anesthesia only blocks sensation to a small area, usually an area of the skin for a short period of time. General anesthesia is most commonly used for surgeries where as local anesthesia may be used for a minor surgery such a small skin tumor removal, or it may also be used in conjunction with general anesthesia to help block pain sensation even further in areas of the body more sensitive to pain, such as with a declaw.
No anesthesia is without risks - local anesthesia can potentially result in permanent nerve damage, and general anesthesia can result in a variety of complications with the most severe being death. There is no such thing as safe anesthesia, but with anesthesia 'accessories' such as presurgical bloodwork, pre-medicants to lower the amount of anesthesia needed and in-depth monitoring during anesthesia, we can reduce many of the risks of anesthesia.
To learn more about anesthesia, keep reading!