Interacting with the cart is most probably one of the single most important thing shop implementors will want to do: adding products to it, changing quantities...
There are roughly two different ways to interact with the cart: through Ajax, or with more standard post-and-refresh behavior.
The normal form POST method is pretty straightforward - you simply POST to the cart’s update URL (shop/cart/update/ by default) and pass it parameters as: update_item-<item id>=<new quantity> Items corresponding to the ID will be update with the new quantity
Posting to shop/cart/delete empties the cart (the cart object is the same, but all cartitems are removed from it)