As far as I understood it correctly:
airspeed: is obtained by measuring the difference in airpressure (using Pitot tubes I guess?) around the aircraft and the airpressure caused by propulsion.
ground speed: is the speed of the aircraft relative to the (immobile) ground.
Ok, I see that there are two types of speed. But why do we need both? Why not just one?
Airspeed is critical for pilots, because it is what the airplane "feels" when it's flying. Too little airspeed and the wings won't generate enough lift to fly, too much and the plane can be damaged. (And you're right, the pitot tube, along with the "static port" is what allows airspeed to be measured.)
Groundspeed is how quickly you're getting where you're going. This is also important, because the engine (or engines) consumes an certain amount of fuel per hour. If you are flying between two airports that are 400 miles apart in a plane that goes 100MPH, with no wind your trip will take 4 hours. If you have a 20MPH headwind your airspeed will still be 100MPH but your groundspeed will be reduced to 80MPH, and your trip will take 5 hours instead of 4. If you've only got 4.5 hours worth of fuel, you're going to need to make a stop. It's really good to know this before you leave!
So airspeed is how fast the plane is flying through the air and groundspeed is, as you observed, how fast you're getting where you're going. Both are important!
Keep in mind there are several types of airspeed.
Indicated Airspeed (IAS) is Airspeed based upon the ratio of stagnation vs static ambient air pressure where the airplane is flying.
Calibrated Airspeed (CAS) is Indicated Airspeed corrected for installation and equipment error. It reflects a more precise airspeed at STP based upon the installation.
True Airspeed (TAS) is the indicated Airspeed corrected for nonstandard pressure and temperature. It reflects the actual speed of the relative airflow over the aircraft. In still air, the TAS is equal to the groundspeed.
Groundspeed is the speed the aircraft is crossing over the ground at. It will be equal to the sum of the true airspeed of the aircraft and the headwind component of the wind aloft at the altitude the airplane is flying at.
It becomes important to differentiate and understand these terms as they will play an integral role in your time distance and fuel calculations in order to know whether you can safely complete a flight. Flight planning does not use linear range as a metric so much as it relies on total available time aloft. Fuel tanks carry known quantities of fuel and engines burn fuel at specific rates at specific power settings at specific altitudes resulting in specific true air speeds during the climb, cruise and descent portions of the flight. Divide the total fuel on board by the rate of fuel consumption, and you know your total flight time available. You will then need to know the winds aloft information to calculate your forecasted groundspeed in order to know if you can safely complete the flight.
Fail to perform these calculations correctly, and you could face a forced landing or a ditching from fuel starvation. If upper level winds are stronger than forecast on a fuel critical flight i.e. one that's planned to be very close on fuel aboard, you will need to be able to make these calculations en route to know whether you will need to divert to an alternate airport or you can press on to the planned destination. This happens a lot both with small aircraft as with this SR-22 which ran out of fuel due to strong headwinds while on a ferry flight to Hawaii and even in large jets. Bangor International in Maine is a alternate for aircraft crossing the Atlantic who get into trouble as is Hawaii and Samoa for Pacific Rim routes.
KIAS (knots indicated airspeed) is the speed which is indicated in the cockpit. It's the aircraft's relative speed through the surrounding air. This speed is crucial for the flying performance of an aircraft because it's the speed which is used to describe the minimum speed and similar important airspeeds.
GS (groundspeed) describes the aircraft's speed relative to the ground. So if an airplane has a groundspeed of 100kts it will fly 100nm per hour relative to the ground. This speed is influenced by the head- and tailwind; the GS will be higher than the IAS if the aircraft experienced tailwind and vice versa.
To calculate time needed to fly a fixed distance you need the GS, as indicated airspeed also describes the speed of the air moving towards you (that means if you are standing on the ground experiencing 40kts headwind, the speed indicator indicates 40kts (as it's calibrated to IAS)).
