A liqueur is one sort of alcohol with a flavouring and/or an addition (Baileys with coffee and cream, Tia Maria with coffee) which is already in it when you buy it)
A cocktail is a mixture of several alcoholic drinks, sometimes with other things added (fruit syrups, chopped fruit, cucumber, mint leaves, little umbrellas stuck into the top, and so on) which is mixed on the spot by someone who has the recipe (or who makes it up as he goes along, with varying degrees of success)
The result of several different cocktails is usually a splitting headache and a rebellious stomach.