Index Rebuild will drop the existing Index and it will recreates the index.
ALTER INDEX index_name ON databasename.tablename REBUILD
Index Reorganize physically reorganizes the leaf nodes of the index.
ALTER INDEX index_name ON databasename.tablename REORGANIZE
Index should be rebuild when index fragmentation is most of the time great than 40%. Index should be reorganized when index fragmentation is between 10% to 40%.
Index rebuilding process uses more CPU and it acquired locking in database resources. In latest SQL Server version and Enterprise version there has option available i.e. ONLINE mode, through which you can be turned on when Index is rebuilt.
ONLINE option will keep index available during the rebuilding.