All foreign visitors to the Republic of Korea must have a valid passport and a visa obtained prior to arrival. Some 99 countries (Including the US, Canada, the UK, Australia and many Western European Nations) have visa waiver agreements with Korea. Citizens of these nations can obtain a tourist visa upon arrival at the airport. Depending on the visit, these may last for 30, 60 or 90 days, or in case of Canada, up to 180 days.
To extend your visa you need to visit your local Immigration Office, at least one day before your visa expire. You will need a recent passport-sized color photo and the application fee. Visa extensions are usually for 90 days. All visas are single entry only, to get a multiple entry visa, you need to apply at your local Immigration office. This can also be handled at Incheon International Airport.
If you'd like to work in Korea, for example teaching English, you need a work visa. These must be obtained from an embassy or visa outside of Korea in your home country. Work visas are usually valid for one year. To get a valid work visa you need an endorsement from a potential employer in Korea. Be aware of the consequences of engaging in money-making business in Korea without a valid work visa can get you fined and/or deported. Also note, that work visas are usually only valid for that specific place of employment. To change jobs, you will need to leave the country and get a new visa.