No, the Cold War is not still going on today. The Cold War officially ended with the fall of the Soviet Union on December 25, 1991. However, for all intensive purposes, the Cold War came to an end with the fall of the Berlin Wall in October 1989.
During the previous 40 years, there was a global stalemate between the Democratic West (United States and Nato), and the Communist Eastern Bloc (Soviet Union and Eastern Europe).
The global threats posed by communism declined after the collapse of the Soviet Union. The country of Russia no longer has the global power that the USSR held for over 4 decades following WWII. The United States is now considered the only global Superpower - so there is no Cold War going on.