Rotary engines (think Mazda RX7/8 Wankel) are somewhat popular in custom gliders and other lightweight aircraft because of their power to weight ratio , compactness/simplicity, and reliability when properly maintained
Typical car engine isn't really optimized for the same requirements of aircraft: more extreme emphasis on power to weight ratio, reliability, and also different powerband
Something like a typical Cessna probably has the closest thing resembling a "car" engine in that it's a similar style inline reciprocating piston engine but special built for purposes of aviation
Old aircraft often used "radial" engines which were reciprocating piston engines but with the cylinders arranged radially.
Almost any plane you fly on commercially today will be using some form of turbine ("jet") engine, even if it has a propeller- turboprops are the most common for shorter distance regional flights, which is essentially a turbine engine powering a propeller.
Larger planes like your 737s use turbofans, which are similar in concept with a turbine engine powering a "fan" but have the fan internal to the engine nacelle and thus can feed the bypass air back into the exhaust stream to gain more efficiency. The large fan in front of the engine is what's producing the vast majority of the thrust. This style engine is generally incapable of breaking the sound barrier, but they can be very efficient at cruising just below it
Turbojet engines (that are propelled by the engine thrust alone and what the layperson might think of as a "jet" engine) would be those found in military aircraft like fighter jets, which may also incorporate an afterburner which is just a way to throw fuel economy out the window for a bit of extra thrust by dumping extra fuel in the chamber downstream of the exhaust turbine. Since the velocity of the exhaust is so high they can be capable of speeds far above sound barrier