I'm afraid that's a generic question for an issue that has specifics.
Some miners you only hook up to a USB port (StickMiners) and (after software set-up) off they go. Connection can be to a desktop computer, a laptop, or (as an example) a Raspberry Pi (a small computing device).
Other miners you have to hook up to a power supply and USB or (more commonly) Ethernet. That ethernet connection either going to a desktop computer/laptop/rpi (with appropriate software) or - if the miner basically has its own small computing device and software on-board - straight to a router/modem.
In terms of connecting to each other - most of them don't. They all work individually. There are some designs that 'string' together - mostly for the benefit of needing less hardware (e.g. one computing device to control several) - which is often just a matter of putting cables between them and making sure each miner knows which one in the string it is.
Some miners you can only get feedback from through common mining software, other miners offer a complete web interface that you can browse to.
If you want to see some specifics of how some miners are connected to each other / computers / the internet / the user, I'd say check out Dogie's guides:
http://www.dogiecoin.com/