A Public synonyms can be accessed by any user on the database.
Synonyms are used to :
Mask the real name and owner of an object. Provide public access to an object Provide location transparency for tables,views or program units of a remote database. Simplify the SQL statements for database users.
An Index is an optional structure associated with a table to have direct access to rows,which can be created to increase the performance of data retrieval. Index can be created on one or more columns of a table.
Indexes are automatically maintained and used by ORACLE. Changes to table data are automatically incorporated into all relevant indexes.
Clusters are groups of one or more tables physically stores together to share common columns and are often used together.