Rather interesting question. It’s not like your car engine, but it does begin with a starter (which can be driven by either compressed air or electricity – depends on the type of aircraft). Generally, the smaller jets and turboprops use electric starters to begin the shaft rotation. The transport category aircraft mostly use air turbine starters which use compressed air.
The start function involves turning the shaft that the compressor is mounted on (and, of course the turbomachinery attached to that compressor by that shaft) up to a rotational speed of approximately 41–62% of maximum RPM (varies for each engine). This speed requires a very “staunch” starter because it remains engaged for a much longer duration than the duty-cycle seen in your car’s starter. Once the shaft reaches the correct RPM (referred to as a “percent of N1” on most jet engines), the Fuel Control Unit (FCU) begins to introduce fuel into the burner cans (or combustion chamber) where high energy “igniters” (similar to a spark plug, but much more powerful) light it off. The initial fuel flow is very low, but increases according to a pre-pre programmed flow schedule built into the FCU which is monitoring the shaft RPM.
The interesting part here is that if the starter were to disengage at light-off, the RPM would stop increasing and the the engine would “temp out”. To protect the engine from this phenomenon, the starter remains engaged and continues to assist the rotating shaft to increase its RPM. This is referred to as “bootstrapping” and it continues until the rotational speed reaches “Ground Idle” – the power setting where the engine can be self-sustaining and run without the assistance of the starter – usually around 70% N1 (70% of the maximum speed of this shaft in normal operation). At this point, the Igniters can also stop (pilot has option to continue using them if entering icing conditions). On smaller engines this process can last approximately 15–25 seconds, but on large high bypass fans, it can take 40–45 seconds or more.