USB-A (officially USB Type-A) is one of the two original connectors defined in the USB 1 standard. It uses a rectangular connector with all its pins on one side, so it will only connect to a port in one way — everyone has had to flip a USB-A plug multiple times to get it aligned correctly.
USB was created to simplify computer peripherals by providing one straightforward connection for accessories. Before it came around, plugging a device into your computer meant dealing with a mess of different cables and port compatibility. Early versions of USB designated one device as the host (usually the computer) and the second as the connected device; they weren't reversible. This is why you don't see USB-A to USB-A cables; plugging one of those into two computers would cause problems when both try to act as the host.
USB-B was the companion connector type in the early days, used for peripherals. You'll occasionally still see USB-B used for devices like printers, but it's much less common now. Mini-B and Micro-B (usually called Mini-USB and Micro-USB) served the same purpose, but were smaller so they could fit on mobile devices like MP3 players, phones and game controllers. These two were much more frequently used than standard USB-B until USB-C started replacing them.
USB-A is still used today and has wide compatibility across devices given how long it's been around. You'll see it on desktops and laptops, routers, game consoles, cars and device chargers (including various charging spots like clock radios, plane seats and couches).