The main Korean language test for non-native speakers of Korean is the Test of Proficiency in Korean (한국어능력시험, 韓國語能力試驗, Hangugeo Neungnyeok Siheom, TOPIK). Not to be confused with the KBS 한국어능력시험, which is a language test for native Korean speakers, TOPIK is administered by the National Institute for International Education (국립국제교육원, 國立國際敎育院, Gungnim Gukje Gyoyugwon, NIIED). The test previously was divided into four sections: vocabulary and grammar, writing, listening, and reading and was offered in standard (S-TOPIK) and business (B-TOPIK) versions. In B-TOPIK, each section was worth 100 points and the totals were simply added together to give a score out of 400 possible points. The S-TOPIK version had three different versions: beginner (초급, 初級, chogeup), intermediate (중급, 中級, junggeup), and advanced (고급, 高級, gogeup). Within the beginner test it was possible to pass level 1 or 2, levels 3 or 4 in intermediate, and 5 or 6 in advanced based on both the average score and the minimum score for each section.
The test changed dramatically in July 2014. The beginner level test is now called TOPIK I with levels 1 and 2 and the intermediate and advanced tests have been merged into TOPIK II with levels 3 to 6. TOPIK I has been simplified to reading and listening sections worth 100 points each and consists of mostly multiple-choice questions. The listening section is 40 minutes long with 30 questions and reading is 60 minutes long with 40 questions. TOPIK II has reading, listening, and writing sections worth 100 points each and has notably faster audio for the listening. Listening section is 60 minutes with 50 questions, writing is 50 minutes with 4 short-answer questions, and reading is 70 minutes long with 50 questions. TOPIK I levels 1 and 2 correspond to Common European Framework of Reference for Languages (CEFR) A1 and A2, respectively. TOPIK II levels 3 and 4 correspond to CEFR levels B1 and B2, and levels 5 and 6 with C1 and C2, respectively. The results of the test you take are valid for two years.
As of this writing, the test is offered six times annually within Korea and slightly less frequently outside of Korea and is continent specific. Please check the official website for current test dates and other relevant information. The website has also provided the public with old tests and the matching audio files so you can better prepare yourself by knowing what the test will be like. After summer 2017, they stopped uploading old exams and require users to login before they can download the exams. If you are unable to download the files from the official site, a Google search will help you find the exams hosted on other websites.
For more information on the TOPIK, including evaluation criteria, you might want to check out the TOPIK page on Wikipedia.