I noticed that some aircraft carriers have an upward starting runway while others don't. What's the reason for this? What are the advantages of both designs and how come not one of them has prevailed over time?
The upward slope on the runway is also known as a "ski jump", and it is to give VSTOL (Vertical or Short Take Off and Landing) aircraft get aloft on the short carrier deck. Carriers without this use aircraft catapults to quickly get airplanes up to flying speed.
The advantages of the ski-jump are its simplicity, which means the carrier has lower build and operating costs. The drawbacks are that VSTOL airplanes are generally more limited in their capabilities, the airplanes that are launched by catapults pack a lot more punch.
There are two basic designs to large deck aircraft carriers operating conventional fixed wing aircraft these days. The first is the short take off but arrested landing (STOBAR) and the catapult assisted take off but arrested landing (CATOBAR) designs.
The STOBAR designs, such as the Russian Navy's Admiral Kuznetsov utilize a ski jump or an upward sloping flight deck at the bow of the ship to launch airplanes via a conventional short take off. The ski jump design offers an increase in vertical velocity to the aircraft just prior to take off giving it more time in the air as it goes off the bow to build up more endspeed to fly. Jets start their takeoff rolls either at launch stations at the bow or the ship's waist equipped with jet blast deflectors (JBD) and hydraulic retractable chocks on the main landing gear to act as holdbacks so the plane can go to full power prior to the takeoff roll.
CATOBAR designs don't utilize a ski jump for an assist, but instead make use of an aircraft catapult either in the form of a steam powered design and also in the form of a large electromagnetic linear motor on newer ships call electromagnetic aircraft launch systems (EMALS). Both systems essentially boost the jet to its required endspeed over launch track of about 100 or so yards. Modern CATOBAR designs like the US Navy's Nimitz or Ford-class carriers have four such catapults, 2 on the ships bow and 2 on the waist, allowing for up to four aircraft to launch at once. Each cat is equipped with a JBD and makes use of a frangible holdback attached to the nose landing gear. The catapult connects to the jet via an above deck shuttle which attaches to the jet either by means of a wire rope bridle or a steel launch bar on the plane's nose wheel.
Both CATOBAR and STOBAR designs recover aircraft in the same manner. The ship is designed with an angled landing area and makes use of hydraulic arresting gear in which an airplane will catch using a steel tailhook to stop in a short space.
STOBAR designs offer a much simpler system with less moving parts requiring maintenance than CATOBAR designs. Conversely CATOBAR designs can launch aircraft with much higher useful loads (fuel and weapons) as well as offering the capability to launch over twice as many aircraft at once as compared with the STOBAR system.