Exquisite: Very good, high quality, and special. Example: An exquisite meal would be a very high quality, usually expensive meal at a fancy restaurant.
Elaborate: Many small details, and usually very advanced beauty or system. Example: The carvings in the stone were elaborate. That means the sculpture had many details and was probably crafted by a skilled sculptor.
However, elaborate can also describe things like: "He had an elaborate way of speaking.", which means he spoke very well and descriptive.
Intricate: Similar to elaborate. It does not focus on beauty as much, but intricate focuses more on small, busy details that are often viewed as complicated, hard to understand, or hard to see. Example: "An intricate machine" means the machine is very advanced, and it has so many small details that it would be hard to understand how it works.