The SR-71, fashioned in different types during it's commissioning, was a spy plane which succeeded the Lockheed U-2, and by all standards, it's capabilities as well. The SR-71 holds altitude and speed records, flying a max 85,069 feet above the earth's surface, and reaching speeds of Mach 3.3 (2,193.2 mph). It achieved these altitudes and speeds without the use of a rocket engine, quite unparalleled at the time.
The engine, a Pratt & Whitney J58, was a jet engine considerably more powerful than others of the time, able to produce a static thrust of 32,500 lbf (pound force), equal to 144,567.2 N (newtons). It was a hybrid between the turbojet and a ramjet, incorporating a much more potent version of the common fanjet (widely used by different kinds of planes, from small jets to commercial aircraft) and the ramjet, an engine which compressed air without a rotary compressor but otherwise operated similarly. The combination of the two allowed the engine to fly at speeds up to Mach 3.3, starting at slower speeds by compressing the air into a chamber where the fuel combustion would take place and power the engine mostly. Once the plane hit higher speeds, the turbojet provided less and less thrust. The power would instead come from compressed air entering the shock cones and put in fuel combustion chamber, where additional fuel would be injected into the chamber, a mechanism also known as afterburner.
This system was inefficient for most planes, and therefore used for only short periods of time; however, the SR-71 sustained most of it's high altitude, high speed operations in afterburner, simply because of the unique hybrid system the engine operated under. Obviously, I haven't mentioned one key component: how was the engine able to sustain itself at such high levels of fuel combustion, speeds?
As seen in the diagram, the shock cones used to power the ramjet-style system was also used as a mechanism of cooling the engine; essentially, the cones channeled the air through the engine at subsonic speeds (below Mach 1) allowing the engine to operate as if it were below the speed of sound. This prevented shockwaves from entering the engine and causing the engine to malfunction, overheat, or break down do to the excess of pressure in the system. The same air used to power the plane was also used to "trick" the engine into working, so to speak!