As far as I am concerned, FlightRadar24 reads out transponder responses.
As most engine driven aircrafts have at least a mode C transponder, is there a site that displays such traffic as well?
Yes, and no.
FlightAware (and I believe FlightRadar24) will show General Aviation aircraft (I panned around a bit and found
N802AF, a Pilatus, buzzing around over Bethlehem, Pennsylvania just now).
The catch is most of these systems are fed with ATC data, and generally only pick up flights that are either on an instrument flight plan, or which have picked up flight following & been entered into the ATC system by the controller who set up the strip.
Most light GA aircraft are flying around VFR (transponder code 1200), and don't show up on these flight tracking sites. Similarly if you're assigned a "local" transponder code rather than a system-wide one you typically won't show up on the tracking sites either (this is often the case when you're flying VFR and go through Approach/Departure for Class C or Class B transitions/arrivals).
As more of the fleet is equipped with ADS-B this coverage may improve, as anyone can set up an ADS-B receiver and skim the traffic data. That data is currently only broadcast when a "participating aircraft" (ADS-B Out equipped) is in the area though...
To properly track GA aircraft, one needs to operate professional equipment or an ATC datastream, both which are not available to most websites.
Flight radar 24 shows GA traffic in some occasions, but in general it cannot track these aircraft.
FR24 works primarily on ADS-B, which is a surveillance technology that transmits aircraft information including position. Strictly speaking, ADS-B messages are not responses since they are unsolicited transmissions.
General aviation equipped with only a Mode A/C (/S) transponder will not transmit ADS-B data. Mode A/C radar replies do not contain position information encoded in them so even if the receiver receives these responses, the position cannot be determined from them.
Only if a message is received by 4 receivers which are carefully synchronized in time, it is possible to determine the position that the message was sent from by analyzing the times of arrival at each receiver. The travel distance of the signal to each receiver is different and even though the signal travels at the speed of light, the time difference of arrival is measurable. Using the 4 times of arrival the position can be revealed using a process called multilateration.
FR24 is using this method for aircraft that transmit Mode S replies but for Mode A/C it is difficult. Firstly there is no aircraft identifier the A/C messages. If a lot of aircraft are flying with the same Mode A code (e.g. 1200 VFR) the arrival times quickly become ambiguous. To add to the difficulty there is no difference between the Mode A and Mode C message formats, so you might easily be comparing the time of a Mode A message from one receiver to a Mode C message from another. Professional multilateration systems used for ATC make use of transmission antenna's that interrogate the aircraft transponders (like radar) to remove any ambiguities. This is not allowed for hobby systems.
Additionally FR24 works on 1090 MHz ADS-B messages. In the USA, an additional ADS-B technology is in use on 978 MHz (UAT). So even if GA is ADS-B equipped, in the USA they might not be detected by FR24.
So to properly track GA aircraft, one needs professional equipment or an ATC datastream, both which are not available to most websites.
The FAA provides 5 minute delayed radar track of aircraft, but only of those aircraft that fly on an IFR flight plan are included. Also aircraft owners can opt-out of that system for privacy reasons.
There are some websites that show GA movement. One that I like is the monitoring website for residents living near Amsterdam airport Schiphol. The website is fed by ATC data, with a delay of about 1 minute. It shows all traffic except military.
FlightAware picks up GA traffic, but only if it is on a flight plan (which sometimes includes VFR aircraft receiving flight following) or if the aircraft is ADS-B Out equipped and within their ADS-B coverage areas.
Either way, you're not currently going to get a full picture of the environment from any source - not in the way that ATC does, at least. Public sources aren't going to show the many, many aircraft not squawking unique codes.
You are right @Force the information presented on flightradar24.com are based on transponder outputs and completed with information stored in a database. Unfortunately a "simple" mode C or S is not sufficient for this. Flightradar24 gets its data not from ATC radars but from private suporters who run a receiver for so called ADS-B outputs from the aircraft. These are more advanced digital information then only the squawk, registration/callsign and altitude of an aircraft used by airliners. But more and more private owners use this technology and that's why you can look foreward to track more and more G/A traffic on flightradar24.com.
I know that flightaware.com also uses this data and additionally uses data provided by the FAA (with a five minute delay) as well. But you won't be able to track any aircraft with a VFR flight plan - unless he is ADS-B equipped - becuase in the US these flight plans are only used for SAR.
A little additional remark: If you observe the German airspace at flightradar24.com you'll sometimes find some microlights on the 'scope' - sometimes I think these aircraft are 100kg aircraft and 250kg avionics.