Common Web Parts and Sub parts for SharePoint 2013, 2016 and SharePoint online

From the MSDN the web part is: By using web parts, we can modify the content, appearance, and behaviour of pages of a SharePoint site by using a browser. Web parts are server-side controls that run inside a web part page: they're the building blocks of pages that appear on a SharePoint site.

The following are the list of the common Web Part categories for SharePoint 2013, 2016 and SharePoint Online.

1. Apps: This web is default for all SharePoint versions even for SPO as well, this is web will content associated and created apps on Site collection on the site. Some of list has been created default with all site which is shown on above.
2. Blog: Provides Web Parts for a blog site.

• Blog Archives: Provides quick links to older blog posts.
• Blog Notifications: Provides quick links to register for blog posts notifications using Alerts or RSS feed.
• Blog Tools: Provides blog owners and administrators with quick links to common settings pages and content lists for managing a blog site.

3. Business Data: This is a group of web part which will display information about Business information, such as status, indicators, and other business data. This group also includes Web Parts for embedding Excel and Visio documents and displaying data from Business Connectivity Services (BCS; a component of SharePoint that allows you to connect to data stored outside SharePoint). This includes many of sub web parts those web parts details are below:

• Business Data Action: Displays a list of actions from Business Data Connectivity.
• Business Data Connectivity Filter: Filters the contents of Web Parts using a list of values from the Business Data Connectivity
• Business Data Item: Displays one item from a data source in Business Data Connectivity.
• Business Data Item Builder: Creates a Business Data item from parameters in the query string and provides it to other Web Parts
• Business Data List: Displays a list of items from a data source in Business Data Connectivity.
• Business Data Related List: Displays a list of items related to one or more parent items from a data source in Business Data Connectivity
• Excel Web Access: Use the Excel Web Access Web Part to interact with an Excel workbook as a Web page.
• Indicator Detail: Displays the details of a single Status Indicator. Status Indicators display an important measure for an organization and may be obtained from other data sources including SharePoint lists, Excel workbooks, and SQL Server 2005 Analysis Services KPIs.
• Status List: Shows a list of Status Indicators. Status Indicators display important measures for your organization, and show how your organization is performing with respect to your goals.
• Visio Web Access: Enables viewing and refreshing of Visio Web Drawings.

4. Community:  This is a group of Web Parts for the community features of SharePoint, such as membership, joining a community, and information about the community. In addition, there are tools for community administrators.

• About this community: This web part displays the community description and other properties like established date. This web part will work on Community sites or any other site that has Community Features turned on.
• Join: Provides the ability for non-members of a community site to join the community. The button hides itself if the user is already a member. This web part will work on Community sites or any other site that has Community Features turned on.
• My membership: Displays reputation and membership information for the current visitor of a community site. This web part will work on Community sites or any other site that has Community Features turned on.
• Tools: Provides community owners and administrators with quick links to common settings pages and content lists for managing a community site. This web part will work on Community sites or any other site that has Community Features turned on.
• What's happening: Displays the number of members, topics and replies within a community site. This web part will work on Community sites or any other site that has Community Features turned on.

5. Content Rollup: New and powerful Search offered with the Continuous Crawl, we can really use it in our operational sites. Such as rolling up search results, providing project summaries, displaying timelines, and showing relevant documents from throughout the site.

• Categories: Displays categories from the Site Directory.
• Content Search: Content Search Web Part will allow you to show items that are results of a search query you specify.  
• Project Summary: Displays information about a project in an easy to read overview.
• Relevant Documents: Displays documents that are relevant to the current user.
• RSS Viewer: Display an RSS Feed
• Site Aggregator: Displays sites of your choice.
• Sites in Category: Displays sites from the Site Directory within a specific category.
• Term Property: Displays the specified property of a Term.
• Timeline: Use this timeline to show a high level view of data from another webpart or tasks list.
• WSRP Viewer: Displays portlets from web sites using WSRP 1.1.
• XML Viewer: Transforms XML data using XSL and shows the results.

6. Document Sets:  Designed for working with sets of documents.

These web part contents below sub parts:

• Document Set Contents: Displays the contents of the Document Set.
• Document Set Properties: Displays the properties of the Document Set.

7. Filters: Web Parts that can be used to filter information. These Web Parts are designed to be connected with other Web Parts in order to provide a useful filtering mechanism. These web part is also having multiple sub parts.

• Apply Filters Button: Add this button to a page so users can decide when to apply their filter choices. Otherwise, each filter is applied when its value is changed.
• Choice Filter: Filters the contents of Web Parts using a list of values entered by the page author.
• Current User Filter: Filters the contents of Web Parts by using properties of the current user.
• Date Filter: Filter the contents of Web Parts by allowing users to enter or pick a date.
• Page Field Filter: Filters the contents of Web Parts using information about the current page.
• Query String (URL) Filter: Filters the contents of Web Parts using values passed via the query string.
• SharePoint List Filter: Filters the contents of Web Parts by using a list of values.
• SQL Server Analysis Services Filter: Filters the contents of Web Parts using a list of values from SQL Server Analysis Services cubes.
• Text Filter: Filters the contents of Web Parts by allowing users to enter a text value.

8. Forms:  Web Parts that allow you to embed HTML or InfoPath forms in a page.

• HTML Form Web Part: Connects simple form controls to other Web Parts.
• InfoPath Form Web Part: Use this Web Part to display an InfoPath browser-enabled form.

9. Media and Content: Web Parts that display media, such as images, videos, and pages. In addition, there is also a Web Part for displaying Silverlight applications.

• Content Editor: Allows authors to enter rich text content.
• Get started with your site: This web part displays a set of tiles with common SharePoint actions.
• Image Viewer: Displays a specified image.
• Media Web Part: Use to embed media clips (video and audio) in a web page.
• Page Viewer: Displays another Web page on this Web page. The other Web page is presented in an IFrame.
• Picture Library Slideshow Web Part: Use to display a slideshow of images and photos from a picture library.
• Script Editor: Allows authors to insert HTML snippets or scripts.
• Silverlight Web Part: A web part to display a Silverlight application.

10. Search: Provides Web Parts for search functionality, such as the search box for entering a query, search results, and refinement of results.

• Refinement: This webpart helps the users to refine search results
• Search Box: Displays a search box that allows users to search for information.
  Search Navigation: This webpart helps the users to navigate among search verticals
• Search Results: Displays the search results and the properties associated with them.

11. Search-Driven Content: Provides Web Parts that display content based on search. For example, Web Parts that show items matching a certain tag, pages based on a search query, and recently changed items.

• Catalog-Item Reuse: Use this Web Part to reuse or republish the content of an item from a catalog.
• Items Matching a Tag: This Web Part will show items that are tagged with a term.
• When you add it to the page, this Web Part will show items from the current site matching the page navigation context.  
• Pages: This Web Part will show any items that are derived from the Pages content type.  
• Pictures: This Web Part will show any items that are derived from the Picture or Image content type.  
• Popular Items: This Web Part will show items that have been recently viewed by many users.
• Recently Changed Item: This Web Part will show items that have been modified recently. This can help site users track the latest activity on a site or a library.  
• Recommended Items: This Web Part will show content recommendations based on usage patterns for the current page.  
• Videos: This Web Part will show any items that are derived from the Video content type.  It will sort items by number of views.  
• Web Pages: This Web Part will show any items that are derived from the Page content type.  
• Wiki Pages: This Web Part will show any items that are derived from the Wiki Page content type.  

12. Social Collaboration:  Web Parts designed for the social components of SharePoint, such as user contact details, shared note board, tag clouds, and user tasks.

• Contact Details: Displays details about a contact for this page or site.
• Note Board: Enable users to leave short, publicly-view able notes about this page.
• Organisation Browser: This Web Part displays each person in the reporting chain in an interactive view optimised for browsing organisation charts.
• Site Feed: Site Feed contains micro blogging conversations on a group site.
• Site Users: Use the Site Users Web Part to see a list of the site users and their online status.
• Tag Cloud: Displays the most popular subjects being tagged inside your organisation.
• User Tasks: Displays tasks that are assigned to the current user.