Range of effects. If you mean general anesthesia (being completely asleep for surgery), the most common side effect is nausea. The anesthesiologist can give medications to help prevent nausea. Other possible side effects are sore throat and possibly sore muscles. Rare complications include malignant hyperthermia (a hereditary problem) and prolonged anesthesia effects. Most people tolerate anesthesia just fine.
Anesthesia. This is depends on whether its ga, local, nerve block or sedation. Issues such as reaction to drug, nerve damage, respiratory issues, damage to teeth, eye abrasions, death, being on a ventilator, feeling pain, etc.. Can all be complications. Without knowing the specifics this cannot be answered in this short space.