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TeamCity mainly consists of two parts:
Plugins can be written for server and for agent. Aside from that plugins are divided by their types:
Extensions can modify some aspects of TeamCity behavior. There are several extension points on the server and on the agent, for example, it is possible to format stack trace on the web the way you need it or modify build status text, read more.
Plugins can also modify TeamCity web UI. They can provide custom content to the existing pages (again there are several extension points provided for that), or create new pages with their own UI, read more.