Hi LindaaT,
Yes, Hosted PBX Systems like RingCentral, Phone.com, and Virtual PBX are interacted with completely online. When you purchase one these services, you are given log in information to enter on their website through your web browser. Once logged in, you are able to customize a whole set of features. You may enter which real world phone lines you want your calls forwarded to. You may set up a menuing system that a caller hears when dialing your number which will prompt your caller to select which extension they would like to be connected with. You may set up a voicemail box and often you may set up a voicemail box for each extension, etc. I personally find toying with and exploring all these features to be pretty fun.
In general you do not need to have a certain speed internet connection for your hosted PBX solution to work well. Your internet connection will only matter when logging into your account to edit your features and this will only require fast enough internet for comfortable web browsing. You'll never have any software or hardware running in your office that requires a constant internet connection. All calls to your virtual number are routed and forwarded to your phone lines by the company whose service you seek over whatever telecommunication network you receive your phone service over.
I'd like to make the distinction that a hosted PBX service is not a phone service provider and should be thought of as a tool for receiving and routing calls to already existing phonelines. An internet based solution for actual phone service would be Voice Over Internet Protocol or VoiP as it is generally referred to.