Mode S and TCAS
There are significant internal and technical differences between Mode S and Mode A/C. A simplified analysis is as follows:
In Mode A/C the interrogator (what you usually think of as the ground radar site) is "dumb" - it just sprays interrogations into the sky, and it can only ask two questions - A or C. The Mode A/C responder (the transponder in your aircraft) replies with a 12 bit data burst which is either a squawk code (A) or your altitude (C).
In Mode S, the interrogator is much smarter - it squirts out a 56 bit (or sometimes 112 bit) burst of data. As you can probably imagine, this means that there's a whole lot more "questions" that the interrogator can ask. The transponder replies with a 56 or 112 bit response depending on the question, so the two boxes can have a fair old conversation. As mentioned by another poster, 24 of the data bits in the interrogation are usually used to select a specific transponder ID, so that the "questions" are directed to a specific box.
In an Elementary Surveillance Mode S transponder (usually in a light aircraft with simpler avionics) the only real conversation is to ask about squawk code, altutude, and a few other simple parameters, much like an old Mode A/C box. The main benefit of Mode S is that the interrogator can poll a specific aircraft much less often, because it can establish a connection based on unique ID, therefore there is less interference and it is easier for the ground radar to discriminate closely spaced aircraft.
In an Enhanced Surveillance Mode S transponder (usually in a more sophisticated flight deck) the interrogator can ask other questions like heading, airspeed, and a bunch of other useful things that cut down on R/T traffic and improve situational awareness for the controller. Obviously the set of data that can be requested depend on what is technically available on the data bus in your cockpit, but the principle is very powerful.
There are other "data link" possibilities being considered which also use the payload capability of Mode S interrogations and responses, such as clearance uplink, weather and traffic information, and so on.
Which brings us on to TCAS. TCAS puts an interrogator into an aircraft, and it polls other transponders just like a ground radar site.
TCAS 1 simply polls nearby traffic, and builds a model of the other aircraft trajectories. If it identifies a collision risk, it says "Traffic, Traffic" and lets the pilot figure out what to do next. Although most TCAS interrogators tend to be Mode S interrogators, it is quite possible to have an aircraft with a TCAS 1 transmitter, but simple Mode A/C transponders. That is not the case with TCAS 2. In TCAS 2, the TCAS computer will not only poll for nearby traffic, it will also propose an escape manouever if it thinks a collision is likely. To guard against the possibility that the threat aircraft has an identical TCAS computer and is going to come up with the identical escape manouever, all TCAS 2 installations combine the Mode S interrogator with a Mode S transponder. If both aircraft have TCAS 2, they will also both have Mode S transponders, and they use the data link capability of the Mode S interrogator/transponder link to agree that they will each propose a different escape manouever - such as one aircraft climbs while the other descends.
So, to answer the earlier question - if you've got TCAS 2, then you also have a Mode S transponder.
Finally, since it came up on an earlier thread, I should point out that there are backwards compatibility features built into Mode S interrogators and Mode S transponders. Although the Mode S system is quite different to the old Mode A/C system, both systems use the same transmit/receive frequencies. Mode S interrogators can also send out Mode A/C requests, and can process Mode A/C replies, so they intersperse Mode S transactions with Mode A/C transactions and can therefore track older aircraft. Similarly, a Mode S transponder can reply to old-style Mode A/C interrogations. As you might expect, there are some clever tricks so that when a Mode S interrogator and a Mode S transponder have recognised each other's capabilites, the transponder stops replying to the Mode A/C interrogations, otherwise some of the advantages of the Mode S system would be defeated.