What is the difference between Report and Alert on the Duplicate Rule means?

The Report option creates a record in the Duplicate Record Item and Duplicate Record Set objects. There is one record created in the Duplicate Record Set object for the occurrence of the the duplicate and one record for each record found by the duplicate rule in the Duplicate Record Item object.

To use it, you have to create a Report Type that looks at those objects, and there seems to be no way to include any related data from the duplicate records on those reports.

However, they are useful when you do not want the user to be alerted - eg it may be a partner user creating the duplicate, and you just want the internal team to deal with the duplicates.

Here's a way to use the records created in a meaningful way... Create a SOQL Statement similar to this

SELECT Id,Name,FieldA__c, FieldB__c, FieldC__c FROM CustomObject__c where ID in (Select RecordID from DuplicateRecordItem)

That way you can at least return some useful data out of Salesforce for the duplicates that have been found.

When you deal with the duplicates, remember to delete the Duplicate Record Set record - it doesn't seem to delete itself when the duplicate has been dealt with.

To add to JodieM's great answer, you can create a custom report type with the primary object being the object you want to find dupes in, and the secondary object being Duplicate Record Items. That way you can report on fields on the object, rather than just seeing if a record is a dupe or not.