Adverse effects and serious complications are very rare, but the possibility does exist. The risk will vary according to your health beforehand and the seriousness of your operation. In Canada, the chance of a healthy person dying as a result of anesthesia is probably between 1 in 200,000 and 1 in 400,000. This is certainly comparable with the risks of commercial airline travel and much safer than travelling by car.
Nausea is relatively common and is treated with appropriate medication. Sore throat associated with breathing tube placement is also quite common, but usually only lasts a day or so. Of course, there are a number of infrequent, less serious complications, such as pain and bruising at the site of injections, headaches, and dental damage. These are usually easily treatable, get better quickly, and are also quite rare.
Most people who undergo anesthesia are completely comfortable and have no complications.