This is a general overview of how a carburetor works and the function of each circuit. Keep in mind, carburetors come in 1-barrel, 2-barrel, and 4-barrel version from many different manufacturers and the exact components and systems will vary widely.
Fuel enters the carburetor through a fuel inlet. The float regulates how much fuel to let in. When the float bowl has enough fuel the float moves up and shuts off the fuel via the needle valve. As the fuel level drops, the float drops down and allows the needle valve to open.
Fuel enters the carburetor throat through a main discharge passage. Fuel is drawn by the vacuum created in the carburetor bore. The fuel is then mixed with air that is drawn in from the top of the carburetor bore, facilitated by the venturi. When the engine is cold the choke valve is partially closed restricting the air flow. This richens up the air/fuel mixture allowing the engine to start & idle cold.
The throttle valve is used to restrict the air/fuel flow. It starts at a closed position and as it is opened by the throttle linkage the air/fuel mixture is allowed into the intake manifold.
The idle circuit is built into carburetors so that your engine can idle without dying. When the throttle valve is closed when at a stop there is little to no air going into the intake manifold. If it wasn't for the idle circuit, the engine would die.
As you can see in the illustration fuel is fed through the low speed jet and down to the idle screw port using the strong vacuum being created below the throttle valve. Air is introduced and mixed with the idle fuel using some sort of air bleed.
Note that once the throttle valve opens, the idle circuit is bypassed, so be sure you are doing your idle mixture adjustment at idle speed.
When the throttle valve is opened a bit, but the RPM is still low, most carburetors will allow a bit more fuel into the intake via auxiliary holes.
For controlling the amount of fuel pushing through the metering jet some carburetors (mostly Carter and Rochester), use metering rods. The metering rod is used to control the size of the opening. During low RPM the metering rod is down and the large step in the metering rod, blocks much of the opening. Under throttle and at higher RPM, the rod raises and allows more fuel through.
Metering rods are controlled in different ways.
Mechanically - The metering rod is connected through a series of levers and rods to the throttle. Pressing on the gas raises the metering rod.
Vacuum Controlled - The metering rod is connected to a vacuum piston or to a diaphragm. The piston is spring loaded to keep the piston up. When vacuum is applied the piston is pulled down closing off the jet.
Both Mechanically & vacuum controlled as in the illustration below. The metering rod is hooked to the diaphragm pull rod which is being held down by vacuum from the manifold acting on the diaphragm. The pump lifter also passes under the metering rod arm. Either loss of vacuum or full opening of the throttle valve will cause the metering rod to be lifted.
Looped Metering Rod
Another type of metering rod is a solenoid controlled type, or looped. A current passes through a solenoid, which is used to move the rod. The rod position is controlled by fuel needs.
An oxygen sensor in the exhaust stream and a control circuit energize, or deenergize the rod solenoid as needed.
The below illustrates a Carter YF carburetor, other models will work in a similar way.