Probably the most common problem that people have is when their car is parked in a moist environment below degrees, resulting in ice forming on some or all of the windows. To resolve this problem, see this answer.
Another problem that is quite common (this happened to me recently), is if you have a car with a car battery that is in poor condition, then the car may not start. This is because the process of starting the car requires the greatest current draw to get the engine to turn over initially. This is also why the headlights will dim temporarily when you turn the engine on (assuming they were on to begin with).
To avoid this problem, you might consider investing in a trickle charger for use overnight so that you can be sure the battery has a full charge when you come to start it in the morning. If this does not fix your problem as the battery is losing its charge quicker than it is supposed to or it appears at the service that your battery is in poor condition, you should invest in a new battery as soon as you can realistically afford to - especially if you live in a place where it is consistently below zero degrees Celsius at the moment.
If you do not want to splash out on a trickle charger (even though they are not that expensive and a worthwhile investment in most cases), but you know that your car battery is in good condition and therefore does not need to be replaced then you can do a simple jump start and rely on the alternator to maintain charge in the battery whilst the car is in motion. This is what most people will do, including myself, though I did not happen to have a trickle charger handy. Otherwise I would have used it.