It really a group of disciplines associated with spacecrafts and outer space flight. It includes mechanical and aerospace engineering (vehicle and propulsion system design), chemical and nuclear engineering (fuel design), electrical and computer scientists (controls and systems design) and astrophysics and astrodynamics (trajectory and flight path design). So most "rocket scientists" are either engineers or physicists. But let's not forget the trades: all the technicians, machinist, welders, etc, that actually build these vehicles.

Rocket Science is Orbital Mechanics. Orbital Mechanics is the mathematical description of how one body orbits another based on the mass and gravitational attraction each body has for each other and the amount of energy each body has. It also includes the mathematics required to describe where the body will be along the ellipse that describes it's orbit at any given time. It addition, it uses the mathematics required to navigate to any place on the globe, and the mathematics to know what direction any place on earth is facing (relative to it's orbit) at any time of the year. This is so you can know where the wreckage is likely to be if there is a failure. It also involve statistics that are used to determine how likely it is for a particular "journey to orbit" to be successful, depending on the winds in the atmosphere.

I never learned this stuff from a mathematics level, but I worked with "rocket scientists" at one job I had, and learned a lot from them at the concept level.

When a spacecraft is in orbit, the orbit is in a particular plane. (Think of a big piece of paper slicing through the globe of the earth at some angle, going through the center point of the earth.) The orbital path is an ellipse drawn on the paper with the center of the earth at one foci of the ellipse.) While it is possible to change the plane of the orbit when in orbit, it is not practical to do so because it involves a huge amount of energy.

So you need to launch the spacecraft from the right spot on earth, facing in the right direction, so when it achieves orbit, it will be very close to being in the correct orbital plane. and you need to launch at the right time so it will be in the right place along the ellipse, if there are other satellites in that same orbit (examples: the GPS or Iridium satellites.)

Then you have to give the spacecraft enough energy for it it achieve the particular altitude needed and the particular shape of the ellipse that is the orbit. So you have to get it going fast enough for it to be in the right orbit and then you have to shape the orbit to what you want it to be. When you add or subtract energy to an object in orbit, it does not just get higher, it changes the shape of the ellipse that describes the orbit. What shape it changes to depends on the direction in which the thrust is given and where on the ellipse the thrust is given.

Orbits can be nearly circular or they can be very elongated. It just depends on where the thrust is given. The mathematical equations that describe this are very complex. Usually, launch vehicles achieve orbit in a very elongated orbit, then they do another burn of the engine, at just the right time, for just the right amount of time, to make the orbit more circular.

Another element is safety. When a launch vehicle uses all the fuel in one stage it drops off, to reduce the mass that needs to be accelerated by the next stage. You have to calculate where on the earth that piece of hardware is to land. You don't want it falling into a city. Also, on it's way to orbit, things can go wrong that would cause the vehicle to not achieve orbit and you need to know where it will land after it gets to the other side of the earth. And sometimes the vehicle gets off course for various reasons (failures of the steering mechanisms for one example) and if left to continue thrusting, no one knows where it will end up, so launch vehicles are equipped with a self-destruct mechanism that can be set off by range control to prevent additional damage. And you need to know approximately where all those pieces will land if that happens. Not just the location on the sphere, but what will be at that location at that particular time (considering the earth is rotating).

The math equations to help you describe locations on the surface of the earth, and where they will be at any time, and the math to help you calculate where, on the earth) the body will land, is also complex.

So one mission we launched took off from a little island in the Pacific Ocean near the equator. It was flying East, and the math predicted that potential debris might fall too close to some of the islands in Tahiti. So the path of the vehicle needed to turn right and left at various times to eliminate that risk. When that risk was past, the vehicle needed to get back onto the right course for the particular orbit needed.

When ascending through the atmosphere, you need to consider the winds. There are winds of various speeds and directions at various altitudes, that change depending on the weather and time of year. NASA has a mathematical model of the atmosphere around the globe for every day of the year. It is expressed in terms of probabilities of wind speed and direction. So when Rocket Scientists are trying to figure out if a particular path to orbit will work, they run simulations hundreds of times with various specific values for the winds at various heights (according to the probability defined by the model). Then they statistically analyze all the variations of the orbit, the vehicle ends up in, and make adjustments to the path to orbit, until they have a very high probability it will end up in an orbit that is acceptably close to the ideal orbit.

Because of the variations of pathways to orbit, the launch vehicle needs to have enough extra thrust in its rockets to give it plenty of capacity to get back on the right course in the cases where the winds blow it off course. But what if the winds don't blow it off course as much as you planned? Solid fuel rockets do not have an off switch. The excess energy in the boosters will cause the vehicle to be in an orbit with too much energy. To compensate for this, as the boosters are nearing the end of their fuel, the vehicle is caused to navigate in a zig-zag pattern such that enough energy is lost (wasted on course corrections) that the orbit turns out just right when all the solid fuel is expended.

All that description is why you really, really need to respect people who really are rocket scientists. They get all those mathematical calculations correct.

When they are correct, the satellite gets into geosynchronous orbit so we can watch TV, or it arrives at the right place in the constellation of GPS satellites so our GPS navigation continues to work correctly, or the spacecraft get to the same orbit that Mars is in, at the same time that Mars is in that place of the orbital path (meaning that they hit a moving target by launching months in advance). Oh, and remember orbits depend on the gravity of the bodies, so what affect does the gravity of the various planets have on an interplanetary spacecraft during it's month's long journey to the distant planet. That needs to be calculated and compensated for in the orbital pathway.

If you have read to the end of this explanation, congratulations. Sorry it was so long. Best wishes in your schooling and in your career. Remember, you don't have to be a rocket scientist to work at a company that launches rockets.)