Craftbukkit is a third-party Minecraft mod which allows you to use Minecraft plugins. The mod is available at Bukkit.org and operates as a JAR file which runs the Minecraft server JAR.
After downloading the latest version of Craftbukkit rename the downloaded file to craftbukkit.jar. You can download the development build or the recommended build. Recommended builds are more stable however they are often previous versions of Minecraft which users with updated clients cannot connect to. Unless the recommended build is also updated for the latest version of Minecraft you'll want to run the development build.
Copy the craftbukkit.jar file to the same directory that contains your minecraft_server.jar. This article assumes you already have a running Minecraft server, if not please refer to my article on setting up your very own Minecraft server.
Start your Minecraft server as you usually would, but replace minecraft_server.jar with craftbukkit.jar. You may want to increase the amount of RAM allocated to your server, as Craftbukkit will require more memory, especially with plugins. Craftbukkit will initialize itself and create a configuration file called bukkit.yml. Nothing in this file has to be changed and configuration is fairly basic.
What's next with Bukkit is choosing and installing plugins. Plugins are available on the Bukkit.org site under plugins.bukkit.org. Here you can list plugins by category, and even search for plugins based on keywords. Plugins which haven't been updated recently likely won't work with the latest version of Craftbukkit / Minecraft. On a plugin page you can download the plugin, find information and documentation, and interact with other users.
Downloading and installing a plugin is a fairly simple process. If you get a JAR file, just put that file in the "plugins" folder automatically generated by Craftbukkit. You will need to restart the server for this change to take effect. The plugin will also generate a folder with the same name as the JAR file containing configuration files and other assets required to use the plugin.
If your download is a zip or rar file you must unzip / unrar this file using a program which can handle these extensions. Move these files to your plugin folder and make sure the JAR file is in the root of the plugins folder. Restart your server and the new plugin will be activated.
If you're having trouble finding out how to use the plugin you downloaded, the download page will have a link to the author's resources where you can learn how to use your plugins to the fullest extent.