In this writeup I'm going to discuss how a spacecraft can go from one orbit to another, and hopefully make it as intuitive as possible. Transferring between two orbital trajectories is done by almost every large spacecraft, including the International Space Station on a small scale.
Moving from one orbit to another
When an object is orbiting a planet, it should stay in that specific orbital trajectory forever. In reality, small effects like solar pressure and aerodynamic drag can and do change the trajectories of things like satellites (with very large long term effects), but if we ignore these effects (a good approximation for short timescales) and only take into account gravity from the central planet being orbited, the trajectory never changes. The satellite in question will continue to take the same path, over and over, forever.
But often times staying in one orbit isn't ideal. There are a huge amount of reasons to change an orbit: Some examples are a crewed capsule wanting to re-enter the atmosphere to land, or a Mars lander orbiting Earth that needs to transfer to a hyperbolic escape trajectory and reach Mars.
Crewed spacecraft like this Soyuz capsule must make numerous orbital adjustments in order to reach their destinations - in this case, the International Space Station.
In order to move to another orbit, a spacecraft must give or take away from itself kinetic energy - essentially speed up or slow down. The total energy (potential plus kinetic) of any given orbit is constant along the trajectory, so to move to a new orbit a spacecraft much change its total energy by changing its kinetic energy at some point. This is usually done by firing a rocket thruster in some direction. This fires fast-moving gas particles, which by conservation of momentum applies an equal and opposite momentum change to the spacecraft and pushes it onto a new trajectory. There are also many more creative ways to change an orbit, including things like solar sails (large sheets that reflect photons from the sun, generating thrust) and drag sails (large sheets that increase the drag force on a spacecraft and slow it down with respect to Earth).
In this post I'll be going over primarily transfers from circular/elliptical orbits to other circular/elliptical orbits, leaving out hyperbolas and parabolas for now (although, as you will see, they aren't really excluded in these transfers).
Ideal Hohmann Transfers
The simplest, and often most fuel efficient, orbit transfer is the Hohmann Transfer. Let's see how one works.
Imagine a spacecraft starts off in a perfectly circular orbit. Since the orbit is a circle, the altitude above Earth is always the same at any time, and the speed of the spacecraft is also constant (although the direction of the velocity changes at a constant rate as the object orbits the planet).
A Hohmann transfer gives an efficient (in fact, in this particular case, the most efficient) way to go from this circular orbit to another circular orbit at a different altitude.
This is done by simply speeding up or slowing down somewhere along the first circular orbit. If the spacecraft fires a thruster opposite its velocity direction, it will speed up. This causes it to travel farther away from the Earth (after the thruster turns off) before coming back to the same position. Speeding up the satellite has now placed the spacecraft into an elliptical orbit, with one end of the ellipse (the perigee/periapsis, closest to Earth) at the altitude of the original circular orbit, and the other end (the apogee/apoapsis, furthest from Earth).
That "top" of your new elliptical orbit then becomes the altitude for the final circular orbit. To complete the Hohmann transfer, a spacecraft waits until it reaches the apoapsis, its furthest point away from Earth. Then, the spacecraft fires a thruster to speed up once more, raising the altitude of the periapsis. Once the periapsis altitude equals the apoapsis altitude, the thruster turns off and the spacecraft is now in a new circular orbit at a higher altitude.
An image of the Hohmann transfer described. The two Delta-V marks indicate the two thruster firings and the direction of the acceleration.
In short, a Hohmann transfer involves three orbits: An initial circular orbit, an intermediate elliptical transfer orbit, and a final circular orbit at a different altitude.
By the way, to get to a hyperbolic or parabolic escape trajectory, you just keep firing the thruster on the first burn to speed up the spacecraft until the apoapsis height goes to infinity, turning the trajectory from an elliptical trajectory into a hyperbolic trajectory (with parabolic right in the middle).
For non-circular starting orbits or final orbits, Hohmann-like transfers can still be used. These transfers won't be exactly like ideal Hohmann transfers (since these need an initial and final circular orbit), but they have similar characteristics and are still typically quite efficient.
An example of a transfer similar to a Hohmann transfer is from an initial circular orbit to a higher elliptical orbit. To do this, I can have my spacecraft fire its thruster once to get onto the elliptical transfer orbit from before, then burn again at apoapsis until my desired close-approach point (periapsis) is reached. The end result is that the spacecraft is now sitting in a new elliptical orbit, transferred to from an initial circular orbit.
Example of a Hohmann-like transfer from an elliptical orbit to a much higher elliptical orbit that intercepts the Moon.
Phasing transfers solve the problem of reaching another object in space.
Say your crewed spacecraft wants to reach the International Space Station, and is currently in the exact same orbit as the station. But unfortunately, right now the ISS is on the exact opposite side of Earth. Even though both you and the ISS are in the same orbit, both you and the ISS will always be half an orbit apart.
To solve this issue and actually reach station, a phasing transfer can be used. A phasing orbit is an elliptical orbit, with one end at the height of the circular orbit the spacecraft was initially in. This elliptical orbit has an orbital period (the time it takes to complete one orbit) or slightly longer or shorter than the original circular orbit.
By entering this phasing orbit, the ISS will now catch up to the spacecraft over time. Every time the spacecraft makes a single orbit, the ISS will get just a little bit closer along the original orbit, and the angular separation will decrease. If the transfer is timed right, the spacecraft can exit the phasing orbit (and get back on the circular orbit of the ISS) right as it passes the ISS, resulting in both the station and the spacecraft being in the same place at the same time.
Example of a phasing orbit, in which the spacecraft enters a small elliptical orbit with a shorter period in order to rendezvous with something in the initial orbit.
The nice thing about phasing orbits is they can be done with extremely little fuel. In general, the more fuel you are willing to use, the faster you can reach your intended target. By only using a very short firing of a thruster, a spacecraft can still rendezvous with another spacecraft - it will just take much longer, but the difference in time between the circular orbit period and the phasing orbit period is very small.
The last transfer I'd like to discuss is the Bi-Elliptic transfer, a three burn transfer between two (often circular) orbits that can sometimes be more efficient than Hohmann transfers.
The idea behind a bi-elliptic transfer is to start off just like with the Hohmann transfer, but continue to fire the thruster until the far-point of the transfer elliptical orbit (the apoapsis) is extremely far from the planet. Then, a long time later at apoapsis, the periapsis (close point) of the new elliptical orbit is raised (via the second thruster burn) to the altitude of the intended target orbit. Finally, once the spacecraft reaches the new periapsis, the thruster is fired a third time to drop the apoapsis down and enter the final circular orbit.
Bi-elliptic transfers can actually be more fuel efficient than Hohmann transfers if the final orbit is much higher than the initial orbit. This stems from the fact that as you get further from Earth (or any planet) on an elliptical orbit, your speed decreases (an example of this is the Moon moving several times slower than the space station). A reduced velocity means it is easier to change your velocity vector, essentially making any sort of thruster maneuver far from Earth way more effective than one done, say, in Low-Earth orbit near the station.
Of course, the big downside (other than having to use the thruster three times instead of two) over a Hohmann transfer is the travel time. Since a spacecraft must travel all the way out to the very distant apoapsis of the transfer elliptical orbit (which can be up to hundreds of thousands of kilometers from Earth), the transfer will take a very long time to complete.
Example of a Bi-elliptic transfer. Note that in this case, since the final orbit has a radius quite close to that of the initial orbit, a Hohmann transfer would be more efficient.
Of course, these are really just arbitrary definitions. You can make any sort of trajectory between two orbits you want, as long as the spacecraft has the fuel and thrust to accomplish such a transfer. But, things like Hohmann transfers and phasing maneuvers are quite common in the world of spacecraft.
Let me know if you have any questions, comments, or corrections. I'd be happy to discuss any of the content here with you if you'd like.
Thanks for reading!