Self-hosted ecommerce means running the shop software on hosting you control: your product catalogue, your customer accounts, your order history, your database. The eight shopping carts in this category are open-source PHP applications, and most of them are older than the hosted platforms people compare them against. osCommerce dates from 2000 and Zen Cart forked from it in 2003. PrestaShop, OpenCart, Magento, CubeCart, AbanteCart and Agora Cart followed.
The obvious alternative is a hosted store. Shopify's Basic plan is £25 a month billed monthly in the UK, or £19 billed annually, plus 2% on every order if you use any payment provider other than Shopify Payments. BigCommerce Core is £29 a month billed annually. Both take the patching and the server off your hands, and both have opinions about what your shop is allowed to do. A self-hosted cart costs the hosting, whatever card rates you negotiate directly, and the work of keeping the software current. No percentage of turnover goes anywhere.
That trade suits large or awkward catalogues, trade pricing and customer groups, unusual VAT or shipping rules, and existing shops carrying years of order history that would be painful to move. It suits nobody who has no plan for security updates. And one of these eight, Magento, is a poor fit for shared hosting in any form. Both of those points are dealt with below rather than skated over.