Update, July 12, 2011: See the MSDN version of this article: Deciding Which SharePoint 2010 API to Use.
Microsoft SharePoint 2010 provides three sets of APIs for developing extensions and custom solutions by using SharePoint capabilities: the server-side object model, the client object model, and Web services. Each API comes with a set of advantages and limitations that you will need to understand before you begin planning and developing custom solutions. For example, the server-side object model provides the most extensive set of options for developing customizations, but it can only be used on computers that are running a version of SharePoint 2010. The sections below describe the three sets of APIs and the contexts in which you will want to use them.
Server-side Object Model
The server-side object model provides the most extensive set of options for extending the capabilities of SharePoint 2010. It consists of every API that is documented in a Class Library contained inside the SharePoint 2010 Class Libraries and Web Service References section of the Microsoft SharePoint 2010 Software Development Kit (SDK).
Any application that uses these APIs must be physically deployed on the server. See Setting up the Development Environment for SharePoint 2010 on Windows Vista, Windows 7, and Windows Server 2008 for guidance on and options for setting up a development environment. In terms of security, these APIs usually have a high level of access, but see Sandboxed Solutions for code access security limitations that are related to sandboxed solutions. This set of APIs provides access to the largest number of features and capabilities. In Microsoft SharePoint Foundation 2010 many of the server-side APIs reside in the Microsoft.SharePoint assembly (Microsoft.SharePoint.dll), and in Microsoft SharePoint Server 2010 many of these APIs reside in the Microsoft.Office.Server assembly (Microsoft.Office.Server.dll). However, you must look at the documentation for each type and member in each API to know for sure which assembly to reference.
Client Object Model
The client object model is designed for use in remote client-side solutions that run on computers where SharePoint 2010 has not been installed. See SharePoint 2010 Client Object Model for more information about the client object model and how to use it. The types and members in these APIs are documented in Client Class Library (for code that executes in a .NET managed application or in a Microsoft Silverlight 2.0 application) and ECMAScript Class Library (for code that executes in the browser).
SharePoint 2010 Web Services
The SharePoint 2010 Web services allow you to integrate SharePoint capabilities into code that runs remotely in client-side or server-side applications that run on computers where SharePoint 2010 has not been installed. SharePoint 2010 provides a limited set of REST interfaces for developers who are familiar with that standard. The Web services provide a fuller range of capabilities than the managed client object model, but they do not provide the same performance and design advantages, such as batching and exception handling. When you develop client-side applications, you should use the managed client object model whenever possible.
Setting up the Development Environment for SharePoint 2010 on Windows Vista, Windows 7, and Windows Server 2008
Using the SharePoint Foundation Server-Side Object Model
Server and Site Architecture: Object Model