A consulate was the office of a consul, an official representative of a foreign government in the territory of another country. Consulates were usually located in cities other than the nation's capital, unlike embassies.
During Day 4, Lee Jong took refuge in the Chinese consulate in Los Angeles. Jack Bauer led a team into the building to extract Jong, but the Chinese consul Koo Yin was killed in the process, causing the Chinese to demand Bauer be turned over to them. ("Day 4: 2:00am-3:00am")
Hans Meyer was at the German consulate in Los Angeles during Day 5, and was picked up from there and taken straight to his diplomatic flight. When interrogated by Jack Bauer, Meyer explained that he had been taken straight from the consulate and could not have been given anything by Christopher Henderson. ("Day 5: 2:00am-3:00am")
Charles Logan took Jack Bauer to the Russian consulate in Los Angeles during Day 6, in order to get information about Dmitri Gredenko from Anatoly Markov. However, Jack was captured by the security of the building, and after permission from President Yuri Suvarov, CTU troops invaded the consulate by force to retrieve Bauer. ("Day 6: 6:00pm-7:00pm")
On Day 8, Chloe O'Brian identified Pavel Tokarev as a Russian diplomat, but could not locate him in the Russian consulate in New York. She passed this onto Jack and explained that he could be anywhere. ("Day 8: 12:00pm-1:00pm")