"Council" is a noun describing a group that legislates. "Consul" is a noun referring to a foreign representative. "Counsel" is the most interesting of the three words, since it can be a verb -- to counsel or advise -- or a noun referring to a person -- a legal counsel, a lawyer. It can also be a noun referring to a thing: counsel is advice. These sound-alike homonyms are spelled and defined very differently.
Council is the only word that refers to a group that provides advice or legislation, for example: "The student council met last night," and "The City Council voted to overturn the new street law." It is also the root for councilor, which refers to a member of the council. Be aware: This is another homonym, which sounds the same as counselor.
Consul is the most limited word of the three. It can only be defined as an ambassador or representative. The exception is a historical reference to a consul of ancient Rome, one of two magistrates who held executive power under the Roman Republic.
Counsel, as the University of Denver reference makes clear, is the most versatile of the three words. It can be a verb designating advice, for example: "I counsel you against that idea." As a legal term, it can be a noun describing a lawyer: "Her counsel cautions against her testifying." It can even be a noun representing advice itself: "I will keep my own counsel."