A primary key is not an index, per se --it's a constraint.
The primary key uniquely identifies a row from all the rest - that means the values must be unique. A primary key is typically made of one column, but can be made of more than one - multiple columns are called a composite....
A unique constraint is implemented in MySQL as an index - it guarantees that the same value can not occur more than once in the column(s) it is defined for. A unique constraint/index is redundant on a primary key column, and a primary key could be considered a synonym but with bigger implications. These too support composites...
In MySQL (and SQL Server), there are two types of indexes - clustered and non-clustered. A clustered index is typically associated with the primary key, and automatically created if a primary key is defined in the CREATE TABLE
statement. But it doesn't have to be - it's the most important index to a table, so if it's more optimal to associate with different columns then the change should be reviewed. There can only be one clustered index for a table - the rest are non-clustered indexes. The amount of space you have to define indexes depends on the table engine - 1,000 for MyISAM and 767 for InnoDB. Indexes, clustered an non, are used to speed up data retrieval and their use can be triggered by using the columns in SELECT, JOIN, WHERE and ORDER BY clauses. But they also slow down INSERT/UPDATE/DELETE statements because of maintaining that data.
Full Text indexes are explicitly for Full Text Search (FTS) functionality - no other functionality can make use of them. They are only for columns defined with string based data types.
Mind that indexes are not ANSI - the similarities are thankfully relatively consistent. Oracle doesn't distinguish indexes - they're all the same.