Most students (think they..) want a classic bike. But classic bikes are heavy, have wide tyres and provide the most wind- and road resistance, so consider a city bike. It brings you faster and more easily to your destination.

Classic
Often have 3 or 7 speeds built into the rear hub (internal gears), but older bikes may also be single speed (‘without gears’).
Often has a basket in front and always a rear rack.
Male/female version.
Robust.
Good for transporting heavy objects.
The chain is often in a closed box that protects it very well.
Runs all year, especially with aluminum parts.
Cons:
Quite heavy. The many kilos make it harder to accelerate.
Has elevated ‘seagull’ handle bar. Your riding position is straight upright, resulting in high wind resistance.

City
Internal or external gears. External gears requires regular lubricating, especially in the winter.
Usually has a rear rack, less often a front basket.
Male/female.
All year round, especially with internal gear.
Less heavy.
Narrow wheels, reducing road friction.
Has a flat handle bar, giving you a more forward bent position. Means better power transfer and less wind resistance.  

BMX / Mountain
Are meant for riding in the woods or on tracks or even off road, but are also fun to ride in the city.
Wide tires that grip well to the ground.
Have multiple, external speeds, typically 18-21.
Riding position forward, meant for power transfer.
Has disc brakes or rim brakes.
Unisex.

Single speed
Some classical bikes are single speed, especially older ones, but this category is more thought of as:
Lightweight
Flat handle bar
Narrow tyres
Quite robust
Can also be fixie
Only male version
Signals simplicity and let’s face it, a bit sexy.

Speed bikes (road bikes)
These are REAL bikes. Instruments built for speed and endurance.
Narrow, high-pressure tyres
Multiple gears (at least 18)
Drop handlebar
Lightweight
Very bent riding position, seat higher than handle bar
Cons:
Need good maintenance
Not so useful for the city, really (cobblestones, drains, gaps, holes). Primarily a summer bike, but some manage to use it throughout.
Very few available.
In brief: HOT.

Electric bikes
Not really bikes. Consider getting a car 😉