Here are the various visa options that are available 

E2 Visa Korea - Standard working visa for westernerns teaching English in Korea  

Visa Eligibility: 
Must be a native English speaker (English being your first language).
Have completed either a 3 or 4 year Bachelors Degree from a college or univesity within the following countires: CAN, USA, UK, AUS, NZ, IR, SA.
Must have an employer who is willing to hire/sponsor your stay.
Hold a valid passport from within one of the following countries: CAN, USA, UK, AUS, NZ, IR, SA.
Employers you can work for with an E2 Visa Korea: Private Schools, Public Schools and Language Institutions. 
Issuing Authority : The Korean Department of Immigration and the Korean Department of Consular Affairs (Korean Consulates) 


E1 Visa Korea - Similar to the E2 Visa  

Visa Eligibility: 
Must be a native English speaker (English being your first language).
Have completed either a 3 or 4 year Bachelors Degree from a college or univesity within the following countires: CAN, USA, UK, AUS, NZ, IR, SA.
Must have an employer who is willing to hire/sponsor your stay.
Hold a valid passport from within one of the following countries: CAN, USA, UK, AUS, NZ, IR, SA. 
Employers you can work for with an E1 Visa Korea : Recognized Korean Universities and Colleges. 
Issuing Authority : The Korean Department of Immigration and the Korean Department of Consular Affairs (Korean Consulates) 


E5 Visa Korea - Professional Employee  

Visa Eligibility: 
varies depending on the organization.  
Employers you can work for with an E5 Visa Korea: Public relations firms and corporations.. 
Issuing Authority : The Korean Department of Immigration and the Korean Department of Consular Affairs (Korean Consulates) OR the Korean Minisitry of Foreign Affairs. 


F3 Visa Korea - Direct family visa (family or spouse)  

Visa Eligibility: 
Must be a direct family member or spouse of the person who has obtained a legal working Visa.
Do not intend to work during their time in Korea. 
Employers you can work for with an F3 Visa Korea: None. This Visa is essentially an extended tourist visa. 
Issuing Authority : The Korean Department of Immigration and the Korean Department of Consular Affairs (Korean Consulates) 


F4 Visa Korea - Overseas Koreans (Korean American, Korean Canadian etc)  

Visa Eligibility: 
Overseas Korean Nationals (Kyopo)
Koreans who have changed their citizenship.
Koreans who have immigrated to their current country of citizenship before 1948.
Koreans who were born overseas with another citizenship.
Foreign Nationals. 
Employers you can work for with an F4 Visa Korea: People with an F4 Visa have the same rights and privileges as Korean citizens, therefore people with this visa can obtain employment within any business or organizations 
Issuing Authority : The Korean Department of Immigration and the Korean Department of Consular Affairs (Korean Consulates) 


Additionally, there are a number of other visas available for westerners who wish to travel, study, conduct business, etc within South Korea.  
  
F1 Visa Korea - Korean Family Visa
B2 Visa Korea - Tourist Visa
E7 Visa Korea - General Work Visa
H1 Visa Korea - 12 month working holiday Visa
Korean Heritage Visa - Korean Descendants

  
Alien Registration Cards  

Anyone who wishes to stay in Korea for more than 90 days (with the exception of Canadian citizens on a 6 month tourist visa) is required, by law, to obtain an Alien Registration Card. The school you work for should bring you to the local Immigration office to obtain this identification card.

This card must be provided In case of health emergencies, trouble with the law, etc. All westerners teaching in Korea are expected to carry their Alien Registration Card with them at all times.