Actions and filters are two of the most useful features of the WordPress Plugin API.
An action allows you to write a function that runs everytime a certain event happens in WordPress. For example, actions can allow you to make something happen every time a post is published. WordPress provides hundreds of actions that you can hook into, but you can also create your own actions.
A filter is similar to an action, but instead of simply acting on a WordPress event, a filter allows you to change that event. Filters allow you to re-write core WordPress functions without ever touching the core code.
In this video, Topher, our WordPress teacher, gives you a 5 minute introduction to actions and filters: