The eBay API (application program interface) is a way for third-party software and web sites to communicate directly with eBay - including common tasks such as listing, relisting, leaving feedback, and so on. There is further information here.
Pretty much all management programs avoid the Sell Your Item pages for listing. They either use the eBay API (which they have to pay for on a usage basis, and have their programs "certified") or they "scrape" the normal eBay web pages to automate the process (which is free but against eBay rules). Good products (of either type) do this completely invisibly - shielding the user from the complex interactions underneath.
Developers who use the API should have more reliable products as the API is the official route into eBay, but charge more for their services to recoup their higher development and access costs. Those who do not use the API tend to suffer when eBay changes the HTML format of the site, but can be much cheaper.