I have a little practical experience of this (quite legally obtained, I hasten to add!)
If you were to distill a fermented beverage such as wine, the maximum concentration of alcohol (ethanol or ethyl alcohol to precise) you could obtain would be 78%. This is due to a curious quirk in what is known as the phase equilibrium of alcohol / water mixes which leads to a low point in the boiling curve at this concentration.
To get more pure alcohol you have to add lime or similar to remove the water chemically.
In practice most distillations yield something in the mid 50's% and some other bits come across too to give characteristic flavour.
Distilling grain liquors should not normally give spirits with dangerous components. This is because the carbohydrates in the must are present more or less entirely as simple sugars such as fructose and glucose and these ferment preferentially to ethanol plus a few "higher" alcohols such as propanols and butanols, a touch of methanol and some other odds and sods like ketones - it is these latter that are largely responsible for hangovers.
Distilling fermentation from grapes and berry sources is more of a problem. The pectin in these can ferment to methanol, which is poisonous. One concentrated up by distillation, methanol really can send you blind. This is avoided by careful monitoring the distillation to discard the first bit that distills (and id high in methanol) or by some jiggery pokery with the enzymes in the must to digest the pectin safely.
There are tests - quite simple ones - that can detect the presence of methanol in significant amounts.
Aside from being illegal, distillation is a considerable fire hazard. We still have travelling stills in France where one can take wine or other stuff to be converted to spirits. They are dying out because no new licences to do this are being issued. The distiller takes a fee for each litre distilled and collects the duty on behalf of the government.
It must be said that the stuff that comes out is pretty fierce.
but I too use it for making liqueurs.
On the whole, I wouldn't recommend trying this at home
Edit - sorry, ethanol and water form a constant boiling mixture AT 78°C of 95% alcohol. I've been drinking the fermented stuff again.
So how do you get spirits then? Do you have to add something to the pure alcohol?
- but then it bacame apparent to me just what I was risking both legally and in terms of the danger I was posing to my housemates so I stopped being so stupid, though in my defence I never attempt to sell it or anything like that, and stuck to making home made wine and beer instead.
I won't confess publicly to what a (more gifted) friend and I decided to make during our chemistry A-level, but trust me it was quite a party 
I learn so much here, it's great.