KEY and INDEX are synonyms in MySQL. They mean the same thing. In databases you would use indexes to improve the speed of data retrieval. An index is typically created on columns used in JOIN, WHERE, and ORDER BY clauses.
Imagine you have a table called users and you want to search for all the users which have the last name 'Smith'. Without an index, the database would have to go through all the records of the table: this is slow, because the more records you have in your database, the more work it has to do to find the result. On the other hand, an index will help the database skip quickly to the relevant pages where the 'Smith' records are held. This is very similar to how we, humans, go through a phone book directory to find someone by the last name: We don't start searching through the directory from cover to cover, as long we inserted the information in some order that we can use to skip quickly to the 'S' pages.
Primary keys and unique keys are similar. A primary key is a column, or a combination of columns, that can uniquely identify a row. It is a special case of unique key. A table can have at most one primary key, but more than one unique key. When you specify a unique key on a column, no two distinct rows in a table can have the same value.
Also note that columns defined as primary keys or unique keys are automatically indexed in MySQL.