What exactly is an intake manifold?

From a mechanics point of view you know it when you see it but defining it is slightly more difficult.

I would break the definition up into 3 separate cases. Carburated or fuel injected gasoline powered engine and a diesel engine.

In a fuel injected engine the intake manifold is everything between the throttle body and the intake port/s of the head/s. I'm referring to the heads and ports strangely because the engine may contain many different configurations from a single head containing a single port to many ports spread across two heads. The manifold directs the air that has passed through the throttle plates to the intake ports. This can be the most complicated of the intake manifolds. It will contain the fuel injectors (excluding direct injection). It may contain variable intake runners. It can consist of a single piece (Ford 4.6 liter) or of two pieces that get bolted together, generally referred to as an upper manifold and a lower manifold (GM 3.4 liter). It may be made out of aluminum or plastic (GM 3.4 liter, Ford 4.6 liter, respectively). The manifold may pass engine coolant through it in some aspect and may contain the thermostat. There are too many different configurations to even mention.

In a carburated engine the intake manifold is everything between the carburator and the intake port/s of the head/s. Because the carburator is an old technology the manifolds are far more simple. The manifold directs air mixed with fuel that has passed through the carburator to the intake port/s. It may be made out or cast iron or aluminum. The manifold may pass engine coolant through it in some aspect and may contain the thermostat. There are too many different configurations to even mention.

Because the diesel may or may not have a throttle body it is more difficult to define. If the air filter has no soft connection to the engine then it's everything between the air filter and the heads. If there is a soft connection to the air filter then where the soft connection turns hard to the heads can be considered the intake manifold. Everything else from the gas engines more or less applies.