Why are engines with carburetors hard to start in cold weather?
They are all harder to start in the cold, fuel injected engines included, and need an over-rich mixture for starting.  The rich mixture is required because a lean one is harder for a spark to light off if the fuel/air charge is cold.  

Now, it's still cold outside after the engine warms up, so the air charge is still cold, initially, but on a warm engine it gets heated as it extracts heat from the intake passage, valve and cylinder head on its way into the combustion chamber (and on Lycomings, the carburetor, which is bolted directly to the oil sump).

Carbureted aircraft engines use primer systems instead of chokes, as mentioned in AEhere's answer, because it's undesirable to have that sort of obstruction device in the intake passage. The primer injects raw fuel at the intake port using a syringe type hand pump, most of which settles on the intake passage walls and runs down the tubes a bit, and sits there slowly evaporating, and when the engine cranks, the vapours add to the fuel normally drawn at the carburetor.  It does this for a period of time after the engine starts depending on how much prime you used.

Fuel injected aircraft engines are primed the same way, but since there are injectors already there at the intake port they don't require a separate primer system.  You just turn on the fuel pump with the mixture rich for x seconds and it injects raw fuel just like a primer system.

Carbureted aircraft engines also use a slightly richer than necessary setting in the idle circuit to help them run better when cold.  That's why you get a little rpm rise when pulling idle cutoff.

You can have too little primer fuel when starting, or too much, and it varies widely depending on the engine, its temperature, and the air temperature, so you have to learn how much is enough on each engine type.  A typical protocol is:


On a cold engine in warm weather, you might use 4 strokes of prime. 
On a cold engine in cold weather, you might use 6 to 8 strokes of prime.
On a warm engine in cold weather, you might use 1 or 2 strokes of prime.
On a warm engine in warm weather, or a hot engine you just shut down, you usually don't need any prime, but may use a little half-shot of prime, or, if the carb has an accelerator pump, a short stroke or two of throttle applied while cranking (to avoid fuel running down into the air box since throttle pumping sprays extra fuel from the accelerator pump right at the carb, and if the engine backfires with fuel in the airbox you can have an intake fire).


Different engine models have different starting quirks, and different primers have varying fuel volumes per stroke.  It's something you learn on that engine type, along with whatever other unique characteristics of a particular aircraft you have to be aware of.