The regular way to burn fuel is not to use an afterburner. The afterburner is an option added downstream of a regular engine to increase its thrust temporarily. It is an interesting alternative to fitting the aircraft with a larger engine. However the additional thrust it delivers is very expensive.
The normal way to burn fuel involves a combustion chamber fed by an air compressor. The afterburner is able to work without a compressor because it uses air from the engine upstream.
I'll split my answer in two sections: The regular combustion chamber and the afterburner.
1. Regular combustor, e.g. for a turbofan
There are four steps in the jet engine, each occurring in a different section:
- Air compression in the compressor.
- Air and fuel mixing in the combustion chamber, and mix combustion.
- Hot gas expansion in the turbine to produce power.
- Gas release in the exhaust.
In addition this engine, which is a turbofan, has a large fan to create thrust. A fan is like a propeller. From left to right: fan, compressor, combustion chamber, turbine, exhaust:
CFM56-5 engine, source
Air is compressed in order to store a large quantity to allow more fuel to be burnt. The more fuel burnt per unit of time, the largest the power of the engine.
Air and fuel are injected in a combustion chamber, which can be of several types, in this example this is an annular duct with individual fuel nozzles.
Each fuel nozzle sprays fuel in an area of the chamber. Air from the compressor is directed to this area to create the mix. Combustion occurs in the annular duct.
There are igniters to start the combustion, then gas continuously burn and as the temperature is extremely hot (1,700°C for this engine), combustion is spontaneous. Igniters can then be turned off, maintained active, or set to be reactivated automatically in case of flameout (choice from the crew).
The annular duct is made of liners with hundreds of holes. Air also enters the holes to create swirls in the combustion chamber and improve fuel dilution, and to cool the liners as this section is the hottest of the engine and materials are close to their temperature limit.
CFM56-5 engine, source
Gas produced by the combustion is directed to the turbine stages. The turbine spins, moved by gas velocity. The turbine role is to extract energy from hot gas to rotate the compressor and the fan.
In this engine the fan just pushes air rearwards. This accelerated air creates thrust by reaction (80%). A small amount of thrust is also produced by the core engine when gas is released in the exhaust (20%).
A more complete explanation in this video. Detail of the combustion chamber:
Combustion chamber CFM56-3, source
2. Afterburner (post-combustion)
The afterburner is a second burner, but it works only in the hot flow of a jet engine upstream, it is a complement to this engine.
As shown in Nasa slide provided in OragnicMarble's answer, thrust in aero-engines is created by increasing air momentum, the product of mass by velocity. It doesn't matter if a large mass is accelerated by a small amount, or a small mass is accelerated by a large amount. However accelerating by a small amount if economically more efficient, that's why manufacturers use a larger fan to produce a larger thrust rather the same fan with a higher velocity.
There are cases where an additional thrust is required only at specific times, and where a larger engine is not desirable (weight and size). For them it makes sense to increase air velocity a bit more, and keep the mass unchanged. The afterburner shines in this niche.
The principle is to take the mass of gas after the normal engine exit, and re-heat it to increase its pressure. Energy is again fuel combustion with oxygen still present in the gas. The compressor is not required to increase the mass of air to be burnt. As there is no compressor, the turbine can be removed too. So in effect an afterburner is a very simple solution.
Engine afterburner concept
This additional heat expands gas, increasing its pressure, and a second nozzle converts the additional pressure into velocity.
Afterburners have many cons: Noise, flame and inefficiency (around 5 times more fuel than a regular combustor). They are not used in civil aviation, except on the Concorde and Tu-144, the only supersonic civil aircraft. Post-combustion was used for takeoff and to accelerate to supersonic speed.
Afterburner on Olympus 593 engine (Concorde), source