Web parts are the building blocks for SharePoint pages. Pages sit in a document library and provide a graphical view that can be customized or hand-built depending on need.
Each document library and list, upon provisioning, create a web part representation of itself that can be embedded on a SharePoint page. As such, web parts are an easy way to create a clickable portal page to quickly access document libraries and lists or for simply resurfacing information from within SharePoint structures.
There are also a few dozen pre-created web parts that provide additional functionality, such as embedding comments on a page or retrieving an RSS feed and rendering a nice formatted list out of the feed items. Developers often create custom web parts that do the kind of actions that default web parts lack. This could include custom buttons, views, fields, and using data stored within SharePoint and providing a nicer interface for modifying said data.
By editing a SharePoint page, users can rearrange existing web parts on a page or add new ones and remove existing ones:
Adding a web part by editing the page and then selecting Insert | Web Part provides a list of all supported and installed web parts on the service.
The interface is a bit retro in the sense that it involves a lot of dragging, dropping, and popup menus to finalize the web part settings:
When editing a web part, the WEB PART tool pane (on the right) becomes visible, allowing manipulation of the Web Part Properties. This includes the title text and possible parameters to instruct the web part to act differently.