조선 (朝鮮, Joseon) was the official name of Korea from 1394 to 1897, the de facto name from 1897 to 1910, and again the official name from 1910 to 1945 (though as a Japanese colony). When referring to Korea during these periods, all Koreans use the word 조선 (朝鮮, Joseon).
After Korea was divided in 1945, South Koreans have generally referred to Korea by the name 한국 (韓國, Han'guk), a shortening of South Korea's official name, 대한민국 (大韓民國, Daehanmin'guk, “Republic of Korea”), which is itself based on 대한제국 (大韓帝國, Daehanjeguk, “Empire of Korea”), the official name from 1897 to 1910.
Middle-aged and older speakers in South Korea may sometimes use 조선 (朝鮮, Joseon) to refer to contemporary Korea without any particular connotation. It is not used for contemporary contexts by younger speakers unless with a sarcastic, disparaging connotation (presumably due to association with North Korea and with old dynastic rule when Korea was weak).
North Korea preserves the name 조선 (朝鮮, Joseon) and uses it to refer to Korea in contemporary contexts. Accordingly, North Koreans will refer to South Korea as 남조선 (南朝鮮, Namjoseon, “South Joseon”).
For Koreans in China, 조선 (朝鮮, Joseon) usually refers specifically to North Korea, while South Korea is referred to as 한국 (韓國, Han'guk).