Internet radios simply connect wirelessly to your home broadband router in the same way that other wifi devices (such as your phone, laptop, printer or smart TV) do.
They come with an app that's got thousands of radio stations available thorough it. However those apps vary across different radios, so there might be some stations which are available on one radio which aren't available on another (and vice versa). Further, some radios let you manually add stations which aren't already available on them, while others don't.
Of course all internet radio stations are available via your PC or laptop as well. I spend hours every day listening to vintage British radio comedy. If that sort of content appeals to you as well, checkout these links:
ROK Classic Radio has two channels devoted exclusively to British radio comedy. (Note that the schedules are in GMT, so you need to add an hour for BST). Their Old Time Gold Channel also includes British comedy content (mixed in with American stuff) but they're not currently broadcasting in line with the published schedule (so it's pot luck as to what you find there):
http://rokradio.com/
British Comedy Radio broadcasts similar content (i.e. The Navy Lark, Men from the Ministry, Clitheroe Kid, Round the Horne, etc) but there's no published schedule:
http://www.abacus.fm/british-comedy.html
They've also got a second channel which broadcasts Goon Shows 24 hours per day:
http://www.abacus.fm/goon_show.html
Fabcat also broadcasts 24-hour Goon shows:
http://goons.fabcat.org/
The best-known portal for finding tens of thousands of internet radio stations is probably TuneIn:
https://tunein.com/
Their service is free but ad-supported. (You can pay for a subscription to get rid of the ads).
InternetRadio provides a similar service:
https://www.internet-radio.com/
(There's a free trial period and then it's subscription-based).
However the best thing to do is to use such services to find radio stations you like and then use Google to find the websites of those stations (which will almost certainly have 'Listen live' links). You can then cut out the middleman and hear the stations free of charge.
Anyone can set up their own radio station via Shoutcast though, meaning that there are about 85,000 stations available on that site (without any subscription). They range from large broadcasters right down to one-man stations:
https://www.shoutcast.com/
The main rival to Shoutcast (with over 17,000 stations) is Icecast:
http://dir.xiph.org/
So you shouldn't be stuck for something to listen to now!
[Typed while listening to 'The Men From the Ministry' on ROK British Comedy 2]