APIs, or Application programming interfaces: APIs are what let you do everything that you can do in the SharePoint user interface, like creating new sites or workspaces, uploading photos or documents, and creating tasks and alerts.
ASP.NET: Microsoft's platform for building Web applications. It's used to customize SharePoint.
Child: This term helps show hierarchy within SharePoint, specifically within sites. A child site is a subsite of the top-level, or parent, site.
Farm, or server farm: A group of servers that share the same administrative tools and are part of the same organization or group.
IIS, or Internet Information Services: The Web server that Microsoft uses to host SharePoint. It runs on Windows Server 2003.
Edited to add: IIS doesn't necessarily run on Windows Server 2003. There are different versions of IIS depending on the operating system you have, such as Windows XP or Windows 2008. So for example, IIS 7.0 runs on Windows Server 2008, and IIS 6.0 runs on Windows 2003.
Library: Sites have libraries, which store data. The most common library in SharePoint is the document library, but you can use libraries to store any kind of files or data.
List: A table of data that is stored in a site. Lists help provide data like tasks, discussions and links.
.NET Framework: Versions 2 and 3 of this set of software installs ASP.NET and Windows Workflow Foundation (WF).
Parent: This term helps show hierarchy within SharePoint, specifically within sites. A parent site is the top-level site; it can have one or several children, or subsites.
Site: In SharePoint, sites are the building blocks, where users can share data in lists and libraries or view and edit Web Part pages. They're created for a specific purpose, so for example, a company may create an HR site, an Accounting site and a Project Management site.
Site Collection: A site collection is a group of sites that are related. These sites have the same owner and administration settings. Site collections have a hierarchy; there's always at least one top-level site in a site collection, with subsites underneath. Every Web application contains at least one site collection.
SQL Server 2000 (or later): Microsoft's database management system.
User Interface: The part of SharePoint that you see and interact with. (It's what appears on your monitor screen.)
Web applications: They provide services, as opposed to Web sites, which typically just display information. Web applications are Web sites that run on IIS. They help create a company's information environment. They are hosted on the Web server.
Windows Server 2003 (or later): Microsoft's server operating system.