What does Sport mode do in a car?

YOUR CAR CAME WITH AN OWNER'S MANUAL. READ IT, DUMBASS. 

Sport mode holds the car in lower gear ratios for longer, improving acceleration.

On modern cars it does more: it will typically firm up any adaptive suspension components, reduce the assistance provided by electric power steering to give a bit more feel and also adjust the steering rate so that there are fewer turns of the steering wheel from lock to lock. The engine will switch to a more aggressive tuning profile instead of following the usual profile which places greater emphasis on fuel economy. On my car it also changes the instruments showing in the binnacle, replacing the economy meter with a tachometer and also changes the lighting from electric blue to red; it also makes the throttle pedal more responsive.

Normal mode reverses all of those changes. 

Eco mode changes the engine management to a profile which maximises economy over performance. The throttle pedal is less responsive, top speed is reduced, the transmission changes to higher gears sooner than normal to minimise engine RPM. On my car (Lexus CT200H) it also reduces the air conditioner output and reduces power to heated seats and the heated steering wheel.

Talk to your geometry teacher.  Something you should know. by grade 6 or 9.  Don't worry.  Teachers love explaining things on a blackboard with chalk.  Lots of pictures.

Sport mode is just a program that changes shift points.

It defeats overdrive and holds gears to higher RPMs before shifting thus giving better performance when needed.

It's how ALL cars used to be before the tree huggers and EPA came along

If you like driving in ECO mode you do not need to worry about the horsepower you are getting.

It adds to the price without adding anything to performance

On older cars it only raised the shift points on the transmission. In other words, kept it in the lower gear longer and shifted at a higher rev. Now it also makes the gas pedal more touchy instead of gradual and with active suspension cars also firms up the simulated spring rate. Just find the setting you like best and you can do mpg comparisons between the settings too.