SQL Sever Data Tools ( SSDT)Business Intelligence Development Studio (BIDS) is old IDE which we were using to develop SSIS Packages,SSRS Reports or create SSAS cubes. Now we are using SQL Server Data Tools(SSDT) to perform the same tasks. By using SSDT you can create Data Source, Shared Datasets and reports. You can also deploy your reports to report server by using SSDT.
Report Builder is tool which is mostly used by the business users to create their own reports. By using the report builder they can use Data Sources from Report Server, Create the Report and Publish to Report Server if they have permission. The latest version available for Report Builder is 3.0.
Report Server:As per Microsoft : https://msdn.microsoft.com/en-us/library/ms157231.aspx
A single report server instance is defined by the complete collection of processors and extensions that provide end-to-end processing, from the handling of the initial request to the presentation of a finished report. Through its subcomponents, the report server processes report requests and makes reports available for on-demand access or scheduled distribution.
Functionally, a report server enables report authoring experiences, report rendering, and report delivery experiences for a variety of data sources as well as extensible authentication and authorization schemes.
Report Database:As per Microsoft website: https://msdn.microsoft.com/en-us/library/ms157231.aspx
A report server contains report server databases that store published reports, shared data sources, shared datasets, report parts, shared schedules and subscriptions, report definition source files, model definitions, compiled reports, snapshots, parameters, and other resources. A report server also enables administration experiences for configuring the report server to process report requests, maintain snapshot histories, and manage permissions for reports, data sources, datasets, and subscriptions.
Report Manager:Report Manager is Web based interface that can be used to perform different tasks such as
Administer the Report Server,View the Reports and Manager the Reports etc.