Selenium server standalone jar is a library that provides you the classes of Selenium automation framework.
Just for additional info, selenium-standalone-server.jar is a bundled jar that contains both API and selenium server.
Selenium server is required to run older selenium RC tests or to run WebDriver tests in remote machines through selenium Grid.
Selenium webdriver is a software framework for the web which is used to carry out automation testing of various web-applications against various browsers. This also enables you to use a programming language in creating your test scripts.
In Selenium WebDriver, there is no need for Selenium Server JAR for performing local tests. Only for remote tests using Grid, it is needed.