I am sure that there will be this type of tourist bus operating in Rome, but I have come up with a fruitless search.
Has someone a website for this type of bus.
Thanks.
Richard.

You are thinking of the Archeobus. You can google it and come up with plenty of information. I think the primary website might be www.miracleplayers.org.

Yes, there are.
Some operated by local tourist and travel agencies such as http://www.romecitytours.com/

You may look for (and I would rather prefer) the service provided by local public transport company.
Lines 110 and 110(open top) operates from Termini central station every 30 minutes. Fare is around €13.00 for the hop on/off service and around €.8.00 for the no stop option.
More on http://www.trambus.com/servizituristici.htm
Unfortunately this site is only in italian but gives you an idea about the route and the kind of coach operating the service .

The company operates also the Archeobus service, as dln said above.
http://www.trambus.com/ArcheoBus.htm

Do a search under " Ciao Roma".

Go to google and enter: "archeobus servizi turistici". You'll see Archeobus and next to it click [Translate this page]

As the other posters have said, Archeobus is what you want to take(http://www.mmdtkw.org/VArcheobus.html or telephone 011 39 06 46952343 / hours 9:30 am-4:00 pm). It is fantastic, only costs 7,75 euro per person. You can't miss their brightly colored busses. Their tours are given in bi-lingual English and Italian and they depart daily from the Piazza San Marco (Piazza Venezia) in front of the church, leaving each hour from 9:00 am until 7:00 pm. There are 14 stops along the way including the famous sites of ancient Rome, in addition to the catacombs along the Appian Way. You can get on and off as you please with the one ticket. If you stay on the bus and don't get off until finished, the RT takes about two hours; however, if you get on and off, you can make it a full day adventure. I highly recommend the Archeobus tours!

Richard: If you're interested in the Archeobus tours I've posted a trip-report, at the top of the page type-in archeobus and click-on "Italy".
IMO you might want to rethink the Rome hop on/hop off buses.
If you are traveling with another the 110 would cost 26e. Let's say your 1st stop is the Colosseum. There is so much to see in that area the Forum, Palatine Hill, Capitoline Museums and the Imperial Forums. Afterwards say you wanted to see the Pantheon & Piazza Navona area and the bus stop is nearby at the Piazza Venezia. The Pantheon is only a 10-15min walk away and the bus would just drop you at Largo Argentina saving you an 800m walk .
The Vatican Museums & St Peter's is a day in itself.
But there are days where this bus would be a good deal.
Can you buy tickets onboard this bus? If not that's a hassle going to a ticket office. Is your hotel along this bus route? If not you have to walk, bus/metro or taxi to one of the stops and again at the end of the day.
Planning your onsite time to catch the next bus, walk to the stop and *find it* and wait.
You should consider buying a "Roma Metro-Bus" map (5.5e and easy to follow) in Rome and taking regular buses, the metro, walking and taxis. Regards, Walter