craft.commerce.cart

This template function returns the current users cart, which is an Order model.

You would get cart for the current user like this:

{% set cart = craft.commerce.cart %}

which is the same as:

{% set cart = craft.commerce.getCart() %}

You can now show the current customer their cart, and add things to this cart with forms in your templates.

Once a cart is completed and turned into an order, calling the function again will give your user a fresh new cart.