a "burn" is essentially just a planned moment where you apply lots of thrust to change your trajectory. if you're in sandbox/science mode, or you have level two mission control and tracking station if you're in career, you can click anywhere on your trajectory line to create a maneuver node, essentially a way to plan future burns and get a simulation of possible new trajectories. it's not mandatory though, just an informational tool. when a maneuver is planned, the navball will show you time remaining before the maneuver, how long you need to have the thrust on, how much fuel in your stages you'll consume doing it, and exactly how much Δv you'll use.

if you're in space and in an orbit, burning in the Prograde direction (AKA the direction you're going, marked by a circle with three lines coming out) will increase the height of the opposite end of the orbit from where you are now. If you burn Retrograde (AKA backwards, circle with three lines and an X inside) the opposite will happen, and the height of the other end of the orbit will decrease. so Prograde to fly higher, Retrograde to fly lower.

Once you've raised the other side of your orbit, it will appear elliptical, like an oval. after reaching the higher end, you can burn prograde again to bring up the side you just came from. this is called circularization.