Casein is the slow-digesting milk protein (contrast with fast-digesting Whey Protein). It titrates amino acids into the bloodstream over an extended period and keeps you fuller longer than whey.
Per Mike Israetel (RP Strength), the popular claim that casein must be taken specifically before bed for "magic bed things" is overstated — he walks this back to a situational recommendation: useful before a long gap without food (e.g. overnight), or when you don't want a big, high-volume shake. Made into a pudding, it's a better-tasting, lower-volume pre-bed option than a shake. Greek yogurt, lean meat, and other low-fat/high-protein whole foods work about as well pre-bed — "casein isn't magical, it's just awesome."
See Daily Protein Intake Guidelines for overall daily protein targets and Leucine for the amino acid most tied to triggering muscle protein synthesis.