The manoeuvre: burn 1 leaves the inner circle for the transfer ellipse; at the apogee a second burn circularises onto the outer orbit (Wikipedia, CC0).
Question
You are in a circular orbit and you want a higher one - where do you point the rocket?
Not up. Point up and you merely rise on the far side and fall back; orbits have no steering wheel. An orbit is a bargain - speed traded for height - and the only way to pay is speed in the direction you are already going. Burn prograde once and the orbit re-forms into an ellipse whose far side, the apogee, reaches the height you want; the burn happened at one point, and the rest of the trip is coasting uphill, losing speed the whole way. When apogee arrives, burn prograde again and the near side stretches to the same height: the orbit is now a circle, a higher one. Two short burns and a long coast - that is a Hohmann transfer, and run backwards (twice against the motion) it brings you down. It also explains the chaser's trap: speed up to catch something ahead in the same orbit and you rise into a slower orbit and fall further behind; the way to catch up is to drop lower first, run fast on the inside, and climb back.
The math
GM = 3.986×10⁵ km³/s²; circular orbits v = √(GM/r); ellipses v² = GM·(2/r − 1/a) (vis-viva); r measured from Earth's centre. Parking circle at 400 km: r₁ = 6,771 km, v = 7.67 km/s. Geostationary: r₂ = 42,157 km (35,786 km up), v = 3.07 km/s. Transfer ellipse tangent to both: a = (r₁ + r₂)/2 = 24,464 km. Burn 1 at perigee: 10.07 − 7.67 = 2.40 km/s. Coast uphill to apogee, arriving at 1.62 km/s after π·√(a³/GM) = 5.3 hours. Burn 2 to circularise: 3.07 − 1.62 = 1.46 km/s. Both burns together: 3.86 km/s. To Mars the same recipe opens windows roughly every 26 months, with a coast of ~9 months.
Interactive figure: a ship loops the 400 km parking circle at 7.67 km/s. Burn 1 (+2.40 km/s) commits it to the transfer ellipse whose apogee reaches 35,786 km - a 5.3-hour coast uphill that ends at 1.62 km/s. Burn 2 (+1.46 km/s) circularises at geostationary: 3.07 km/s, less than half the speed it left with; decline to burn and the ellipse keeps orbiting too, back down to 400 km every lap. Drawn to scale: yes, the parking orbit really is that thin a margin above the ground.
Why it matters
Every communications satellite that reached geostationary orbit flew this route; Apollo's trans-lunar injection was its first crewed instance of burn 1; a few times a year a docked cargo ship fires to lift the ISS back up after drag has pulled it down - the same bookkeeping, done without leaving home. Walter Hohmann, a German engineer with a doctorate who published the manoeuvre in a 1925 book and died twelve years before the first satellite flew, named nothing after himself - the space age did. And keep the ledger in view: 3.86 km/s spent, arrival at 3.07 km/s - less than half the speed you left with, in higher ground. Orbit is not steering; it is accounting, and every orbit here balances in Kepler's books.
