Hi,
HikaShop stores the cart of the customer in the database.
If the customer is logged in, the cart is linked to his account, so next time the customer comes, he will see his cart.
If the customer is not logged in, the cart is linked to his php user session (there is a setting in the Joomla configuration for the session duration if you want to change it). So as long as the user session is not destroyed (by either closing his browser, or turning off his computer), the session don't be destroyed and he will be able to see his cart on your website.
All this is basically how most online shops work.
We also have a cart reminder plugin here:
www.hikashop.com/marketplace/product/241-reminder-plugin.html
This plugin allows you to notify the customer a certain duration after the last time the cart was modified by the customer so that they can continue their order and pay. Note that the notification is by email, and thus it requires that the customer also entered his email address (either when starting a guest checkout, or by registering/logging in).
This plugin also handles order reminders:
When the customer clicks on "finish" on your checkout, the cart is converted into an order and the customer is redirected to the payment gateway for the payment. If the payment is not done, the order is not confirmed. This plugin can also notify the customer after a certain period after the order is created so that they can try to pay again for their order.