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Air traffic control radars come in two flavours, with most ATC centers using both and displaying the output of both on a single screen:

• 
  Primary surveillance radar (PSR) is a true radar system, which detects aircraft (and also non-aircraft objects, such as birds, balloons, weather, parts detached from aircraft, the ground...) by bouncing radio waves off them. PSR is valuable because it can detect not just aircraft with functioning and responding transponders, but also aircraft without functioning transponders and the aforementioned non-aircraft-but-potentially-of-interest-to-or-regarding-aircraft objects. However, in order to maximise the amount of airspace covered per 360º sweep, the radar beam from most civilian PSR transmitters is fan-shaped; at any given moment, the beam covers (for all intents and purposes) just a single azimuth, but is spread out over a wide range of elevation angles. The radar has no way to determine the elevation angle of a returning reflected signal, which makes most PSR systems useless for discerning altitude, and introduces some imprecision and a variable amount of inaccuracy into the ground position of a target (as the object producing the return could be located anywhere along a vertical arc segment at the target’s slant distance in the plane of the radar beam).¹
  


• 
  Secondary surveillance radar (SSR) isn’t, strictly-speaking, radar at all. Instead of bouncing radio waves off the local flying-object population, it, in essence, calls out “Is anyone there?” in the blind; transponder-equipped aircraft hear this call, and the aircraft’s transponder then broadcasts the aircraft’s position, altitude, airspeed, direction, flight number, aircraft type, licence, registration, proof of insurance, etc., allowing the SSR system to plot the exact position of the aircraft generating the return, along with the aforementioned other various interesting pieces of information. This makes SSR extremely useful, especially in congested airspace, but it can’t detect anything without a transponder, or with a broken transponder, or with their transponder turned off. SSR is planned to eventually be replaced by ADS-B, which works in essentially exactly the same way, except that the aircraft’s transponder broadcasts all of the aforementioned information in the blind, without being asked, thus eliminating the need to have a large, expensive SSR transmitter for each ATC facility.

Here’s an illustration of the difference in the determination and display of target positional information between primary and secondary radars:

Separating secondary targets from each other is (assuming that the air traffic controllers and pilots concerned are paying attention and doing what they’re supposed to), obviously, trivial; you just need to avoid having two or more aircraft occupy the same position at the same altitude at the same time. Separating primary targets from each other is also fairly trivial, despite the fuzzy positional information and the lack of altitude information; two or more aircraft occupying different vertical arc segments (and, thus, separate from each other) will show up as two or more separate targets, so the trick is to keep two or more aircraft from occupying the same arc segment. If you’re dealing with just primary targets, separation between aircraft can be assured by keeping the dots on the screen away from each other.

If, however, you’re dealing with a mix of primary and secondary targets, I see a problem arising: since positional information is determined differently for primary and secondary targets, an aircraft in a given position could show up in one of two different places on the screen, depending on whether it’s a primary or a secondary target:

It should be immediately obvious that the same effect could be produced if the primary and secondary targets were produced by different aircraft; two aircraft in dangerously close proximity to each other could appear to the controller to be separated by a safe distance, if one of the aircraft is a primary target and the other is a secondary target.

How to solve this problem?

¹: Some PSR systems (primarily military systems, as it’s much easier to vector aircraft to intercept an intruder if you know its altitude and its exact location) do provide altitude information; some of these work by using a narrow “pencil beam”, which allows the determination of a target’s elevation angle at the cost of greatly-increased scan times, while others use a pair of radars at different heights and measure the difference in slant distances to the target between the two radars, allowing the target’s altitude to be determined using trigonometry.

Mode C transponders only transmit a squawk code (same as mode A) and altitude, so the range error due to slant distance is the same for SSR as for PSR, and the two can be easily merged. Mode S adds some other details, but none that are relevant here.

ADSB transmits lat/long (and many other things), so it has no slant range error. However, ADSB messages also contain the same squawk code and altitude as in mode A/C/S, which means ADSB targets can be trivially merged with their corresponding SSR targets.

Typically the surveillance information from multiple sources (PSR, SSR, ADS-B, multilateration, etc) are combined into a single "system track" using a Kalman filter. The ATM system has specific code to deal with crossing tracks where the labels may get confused, and controllers have procedures to fix this when it does happen.

The ATM system does its best to avoid displaying multiple targets for the same aircraft.

I was told that separation with primary targets is usually not assured.

The primary radar output usually contains a lot of ‘ground clutter’ like cars on the motorway and wind turbine rotors—it can filter out reflections of static objects, but these are moving and the system cannot be sure—so finding a relevant primary target in it is quite hard.

So controllers only look for primary targets if they have another indication that an aircraft should be there—either because it is communicating over radio, but does not have transponder or it had failed, or because another pilot already reported an aircraft that is not communicating at all in the area (gliders, balloons and such often don't have radio at all, and you don't have to have it in the uncontrolled airspace).

Note that the precision of an area radar is not that high, maybe a mile, so the slant error is often below the resolution anyway. And since the secondary radar collects transponder responses to the sweeps of the primary radar, it has no problem matching the secondary targets with the corresponding primary ones.

A transponder does not, in fact, transmit the plane's position, or altitude, or direction of flight, or any of the other things you said. The standard civilian transponder only transmits two or three pieces of information:

1. The 4-digit transponder code entered by the pilot.


2. Whether or not the pilot has pressed the "ident" button in the last 12 seconds.


3. The current barometric pressure as measured by the static port (only for Mode C).

Position information is derived from the transponder reply the exact same way as it is for primary radar: The direction is determined based on which way the dish was facing at the time it sent the interrogation, and the distance is based on the time it took for the reply to come back. This means that the slant error is exactly the same for primary radar as it is for the secondary.

Usually the PSR and SSR radars are co-located. A plot combiner at the radar site combines them (or not), and transmits reports to the ATM system as SSR reinforced, SSR only, or PSR only.

Then, expanding somewhat on @Greg Hewgill's answer, the ATM system receives input from multiple radar sites, and possibly other sources such as ADS-B. In most of the developed world except for extremely sparsely populated regions, aircraft are usually within the coverage of multiple radar sites, which may be tens or even hundreds of miles apart. Since the earth is curved, the "horizontal" plane at one radar site is different to that for a radar 200 miles away. The positions must not only be slant-range corrected but also converted to a common "system plane" so that position reports for the same aircraft from different radars can be correlated to the same system track. Occasional missing SSR data from one radar can usually then be obtained from a different radar (except of course in cases such as transponder failure).