I simply know that a single free quark does not exist. What is the reason that we can not get a free quark?
If we can't get a free quark then what is single-top-quark?
A free quark is like the free end of a rubber band. If you want to make the ends of a rubber band free you have to pull them apart, however the farther apart you pull them the more energy you have to put in. If you wanted to make the ends of the rubber band truly free you'd have to make the separation between them infinite, and that would require infinite energy. What actually happens is that the rubber band snaps and you get four ends instead of the two you started with.
Similarly, if you take two quarks and try and pull them apart the force between them is approximately independant of distance, so to pull them apart to infinity would take infinite energy. What actually happens is that at some distance the energy stored in the field between them gets high enough to create more quarks, and in stead of two separated quarks you get two pairs of quarks.
This doesn't happen when you pull apart a proton and electron because the force between them falls according to the inverse square law. The difference between the electron/proton pair and a pair of quarks is that the force between the quarks doesn't fall according to the inverse square law. Instead at sufficiently long distances it becomes roughly constant.
I don't think this is fully understood (it certainly isn't fully understood by me :-), but it's thought to be because the lines of force in the quark-quark field represent virtual gluons, and gluons attract each other. This means the lines of force collect together to form a flux tube. By contrast the electron-proton force is transmitted by virtual photons and photons do not attract each other.
Finally, top quarks are usually produced as a top anti-top pair. It is possible to create a single top quark, but it's always paired with a quark of a different type so you aren't creating a free quark.
John Rennie's answer is good, just a few words to add on 'single top quarks'. The mental image of rubber bands works fine at low energies. Maybe you've heard about mesons or baryons, or simply particles like the pion. These are 'bound states of quarks', i.e. what happens when you break the rubber band and create a new pair of quarks. Then a quark + anti-quark form a pair together.
At accelerators, you can get beyond this description: collision energies at modern accelerators as the LHC are (roughly) 100-1000 times higher than the energy that binds quarks together. So in the middle of a collision you can forget about rubber bands and imagine that quarks behave just like single electrons. But then quarks lose energy (for example due to radiation), so inevitably they'll go to the phase where they form bound states and you're back at the rubber band-description. This process is called hadronisation.
Finally a word on 'colour confinement' (the term people use to explain why free quarks don't exist). Each quark or gluon has a 'color charge' (which is just a quantum number). To get a colour-neutral object, you need at least two quarks in a bound state, but you can have more. (This is just $SU(3)$ group theory.) So if you assume that all physical (asymptotic) states need to have no net colour charge, then you can explain why free quarks don't exists. But even though this hypothesis is true experimentally, there is no mathematical reason for it.
As J. Rennie points out, none of this is completely understood and a full mathematical description of confinement would be the biggest breakthrough in quantum theory in decades.