The following is a an explanation of the functions a the Weber Carburetor, the 34 DMTR.  I have made the description to introduce principles  of the general functioning of Weber carbs.  There are other functions not discussed here, but can be found elsewhere.  I'd recommend the Haynes book. 
The DMTR is a compound progressive carburetor.  Compound because it has two barrels, progressive because the throttles open progressively, not simultaneously.  It has three circuits:  Idle/low speed, primary and secondary.  These three circuits correspond to the motor's operating range:  idle / low speed, partial throttle, and full throttle.
Here's a scan of a photo of the deck of the 34DMTR.  The float bowl cover has been removed, so the float bowl is visible, surrounding the round barrel on the right.  Labels for 3,4,5,6,7,& 8 are in the float  bowl.
 

This first diagram shows the carburetor in cross section; looking from the right.   This is the same orientation as the photo above. The screws on the left side of the image are on the back.  It shows the idle / progression circuits in action.  The full figure is at idle, the close-up at the upper right is a low speed, the close up at lower right is at part-throttle.  If you look at throttle plate in the left side barrel, you will see that the plate is turning clockwise to open the passage.

 

 
Here's what happens when the engine is at idle, the accelerator pedal at its stops.   At idle, the throttle plate is almost, but not entirely closed.  There is a very small gap between the throttle plate and the wall of the carb barrel.  This gap creates a venturi effect, an area of low pressure.  

It's important to understand the venturi effect, since it is the basis of the Weber's functioning.   Effectively, if you move air through a tube, and suddenly that tube gets smaller, the air moves faster through the smaller part, and the pressure drops.  It is the drop in pressure that is important.  If you have a low pressure area and a high pressure area, fluid (in this case gasoline) moves from the high pressure area to the low pressure area.    The gasoline in the carburetor is stored in a bowl that is exposed to atmospheric pressure, 14 pounds per square inch, or 1Bar.   The pressure at the small part of the venturi is less than 1 Bar.  So, the higher air pressure of 1Bar pushes the fuel towards the low pressure area in the venturi.

Anyway, there is a little gap between the throttle plate and the wall of the barrel.  So, the air moves a lot faster at this point, and the pressure drops.   At this gap Weber has located a small hole that is the end point of the idle circuit  (# 39 in the drawing below, with the red arrow)
 The hole at #39 receives a mixture of fuel and air.  Why the mixture of fuel and air?  Fuel by itself would flow too quickly, and would be difficult to regulate.  Mixing the fuel with air slows its passage through the circuit.  Weber calls this fuel 'emulsified'.  Air enters through a hole in the deck of the carburetor (#36 in the drawing immediately above and #4 in the photo at the top ).  The mixture is determined by the screw just to the left of hole #36. It is labled in the top photo as #3.   At the base of the screw is a "jet".  The jet is a machined metal tube, closed at one end with a single hole in the end or the side.  There are slots or holes in the sides of the tube.  The hole is immersed in gasoline, and the gasoline is exposed to the air through the slots in the tube.  This air is at atmospheric pressure.  The fuel is at atmospheric pressure also, as the float bowl is vented to the atmosphere.  The pressure at the venturi is less than atmospheric, so the mixture of fuel and air is pushed through the circuit (37, 41, 45) to the hole in the barrel(#39 in the drawing, #1 in the photo).    The screw # 40 regulates the amount of emulsified fuel that is fed to the engine.  This screw adjusts the idle speed.
 Here's a closer view of the throttle plate area as the accelerator starts opening.  Push on the accelerator, the throttle plate opens a little, and the venturi effect at the lowest hole diminishes.  The area of low pressure moves up in the barrel to the spot of minimum gap between the plate and the barrel wall. 

Sure enough, Weber put a hole there too, to provide a smooth transition in fuel flow as the throttle is opened.   Screw # 44 regulates this amount of emulsified fuel to the motor.

When the throttle is opened further,  the venturi effect at the primary barrel wall diminishes.  There is no more pressure differential at the edge of the primary throttle to pull emulsified fuel into the barrel.  There is enough air moving through the primary barrel to bring fuel through the main venturi (more on this later)   At a certain point, usually when the primary throttle is about 2/3 open, the throttle on the second barrel (#  6 in the photo)  starts to open.  The same procedure happens here as happened on the primary throttle.  A venturi is created at the edge of the throttle plate, and this venturi pulls emulsified fuel through a strategically located hole in the second barrel wall.  This follows passage 48 & 50 in the numbered diagram above.  In the drawing, #39 points to the air supply.  In the photo, #8 points to the jet for this circuit and #9 points to the hole that supplies air to this circuit.  These are analogous to #3 and #4 for the primary circuit
Here's another cross section of the carburetor. I flipped the image to make it less confusing.   Primary barrel on the left and secondary barrel on the right.  It shows the carb at full throttle.
The red is the fuel, the blue is air at atmospheric pressure, purple is emulsified fuel.    Note first that the level of the fuel is even with the top of the venturi (#12 in the drawing).  This is important.  If the venturi were higher, the pressure differential between the venturi and the fuel would be too great.  The fuel would have to be pulled up against gravity.  If the venturi were too low, then fuel would spill out into the intake manifold.   Fuel is mixed with air in a sleeve /  cylindrical chamber #16. This chamber is connected directly to the venturi #12.   A Main Jet (18)

    is immersed in fuel.  It has a hole bored through it to allow a certain amount of fuel to pass. The corona of the main jet blocks fuel from passing into the sleeve.  Fuel can pass only through the hole.    The jet is attached to the end of an Emulsion Tube     .   The emulsion tube is hollow,  with holes machined into the sides.   The machined holes allow a certain amount of fuel/air mixture to pass from the inside outwards to be carried out to the venturi.  At the top of the tube is another removable fitting, the Air Jet  .  In the top photo, the air jet for the primary circuit is #5, the air jet for the secondary circuit is #7.     The special machining of this hole  allows a fixed amount of air to pass.    Fuel comes up into the bottom of the emulsion tube, air comes in from the top of the emulsion tube, the two are mixed inside the tube, and pass through the holes in the side to the venturi.

  The venturi is a tube inside a metal ring.  The metal ring fits inside the carb barrel.  You can see the venturis suspended in the center of the carb barrels in the top photo.  The tube is suspended on a crossbar.  One of the arms of the crossbar is hollow, and is connected to the sleeve holding the emulsion tube.   The emulsified fuel exits the tube through the holes in the side, into the sleeve, and then on out to the crosspiece of the venturi.  The cylinder in the center of the venturi is a low-pressure zone.  This low pressure pulls the emulsified fuel through the passages.  The venturi is designed so that the low-pressure zone exists only when a lot of air is moving through.  At lower air volumes, such as at idle, there is no vacuum differential.
 
I mentioned earlier that I admired the elegance of the  Weber carburetors.   This is a system that provides properly metered fuel at idle, low throttle, part throttle, and full throttle.  The only moving part is the shaft of the throttle plates, and these are moving parts already on the motor.  All the metering and mixing is done by differences in air pressure.  No additional moving parts are necessary to accomplish this task.    Another very nice thing about the Weber design is the range of potential calibrations available.    All the jets, emulsion tubes, air bleeds, and venturis come in various calibrations.  The jets and tubes can be accessed from the top of the carb, and removed with a screwdriver.  Note in the top photo, items #3,5,7 & 8 are cut to accommodate a screwdriver. The carb need not be disassembled to change these parts.    So, the motor can be tuned for a wide range of operating conditions, all with the same carburetor.

The main criticism of Weber carbs is their vulnerability to clogging.  Having dealt with this personally several times, I can understand the critique.  The clogging usually occurs on the primary air jet, (#5 in the photo & #11 in the drawing immediately above)    The primary emulsion tube gets filled with junk, and air & fuel cannot mix and pass to the venturi.  This usually happens with before any clogging of the low-speed/idle circuit.  So, the car idles OK and runs fine at low speeds, but stumbles and dies on acceleration.  It's on acceleration that the primary circuit becomes active.  Engine speed increases, the throttle lets more air in, but the primary circuit is not allowing the proper amount of fuel to pass.  So now, there is not enough fuel to burn the air, and the motor stumbles and stalls.   Delightful in city driving.

Here's how I try to resolve the problem.