It gives you the ability to add products, take orders and manage payments and stock directly in your WordPress administration panel. WordPress themes generally need to be built with WooCommerce support in mind for them to display the store, cart, checkout, and other WooCommerce specific pages correctly. The default WooCommerce theme is Storefront. It can be downloaded for free from WordPress.org.
There are a multitude of plugins built specifically for WooCommerce that extend it in different ways: additional product settings, order status managers, different payment gateways, and so on. Many WooCommerce plugins are premium plugins, but there is a decent selection of free WooCommerce plugins on the WordPress.org plugin directory as well. Like the service WordPress.com, WooCommerce is owned by the company Automattic.