肝 and 胆 refer to different organs.
- 肝臓【かんぞう】: The most common word for "liver".
- 肝【かん】: The same as 肝臓, but medical experts prefer this, especially in writing, because it's shorter.
- 胆嚢【たんのう】: Gallbladder, not liver.
- 胆【たん】: Medical experts prefer this to 胆嚢.
- 肝 • 胆: It's not a single word, but just "liver and gallbladder". These two organs are often paired like this because they form the hepatobiliary system.
As food, liver is called レバー or キモ. You would see them at yakiniku restaurants.
Metaphorically, 肝心【かんじん】 (na-/no-adj) and 肝【きも】 (noun) refers to the most important point of a procedure or information. You can say "肝心なのは80℃以上に温めないことです", "ここが肝ですよ、よく聞いてください", etc. 肝心 is sometimes spelled as 肝腎, too, but I believe 肝心 is more common.