Britishtv.club, which you are already using, is one example of a streaming service. I think it only streams live programmes but I don't know for sure. BBC iPlayer, ITV Hub, All 4 etc are also streaming services but are restricted to UK viewers. They have both live broadcast and some catch-up. You can get round the restriction by using a Virtual Private Network, VPN, that makes your device look as though it is in the UK.
Streaming simply means sending a digital version of the programme to your device over the Internet in real time. Streaming of catch-up programmes simply means that somebody has stored a digital version of a programme that was previously broadcast on a server somewhere. You connect to their server and stream that programme to your TV in the same way that you use Britishtv.club. Catch-up usually only goes back a short time. I use two paid for streaming services, ExpatVision and SavvyTV. The former goes back one week while the latter goes back four weeks but doesn't have catch-up on as many channels as the former. Both services also have box sets of TV series and films on their servers and you can view them in the same way.
The exact way that you access the streaming service varies between suppliers. As far as I can see Britishtv.club works on virtually any device. The two I use only work on a MAG streaming box which costs between £50 and £100 depending on the exact model.
PS SavvyTV is quite unreliable in the evenings at the moment. It looks like it is a bandwidth problem at their servers. I think the company is based in Spain although I don't know if their servers are there. If they are it may be that their bandwidth is being throttled because of high demand.