It depends what kind of product you're talking about.
If it's a TV then there is a circuit board made from scratch. A TV has rather special video processing capabilities that are probably implemented with a special TV chip (or several). At minimum it needs to upsample the video signal to the resolution of the TV screen, and then add a menu on top of that. The computer may be a full computer or a relatively big microcontroller (see below). If it's a smart TV, then it's definitely a full computer (a.k.a. application processor), and the computer may also be doing the video processing with its GPU.
If it's a microwave oven they use a microcontroller which is a little computer in one chip. Microcontrollers are all over the place - they're extremely useful whenever you just need a little bit of computing - like processing the buttons on the microwave and counting down until it stops. Nowadays, pretty much anything that has buttons has a microcontroller - including your computer keyboard and mouse.
As a hobbyist for some reason it seems to be rather difficult to make a computer board like the Raspberry Pi (though it is possible (though not using the same chips on the Raspberry Pi because they are specially made for the Raspberry Pi)). Your recommendation is to use something like a Raspberry Pi because it is easy - you don't have to mess around with DRAM timing and custom bootloaders and stuff like that. Very many microcontrollers are also hobbyist-friendly.