How are microscopic transistors on microchips made?
Microchips are made using a very wide variety of process steps.  There are basically two main components to each step - masking off areas to operate on, and then performing some operation on those areas.  The masking step can be done with several different techniques.  The most common is called photolithography.  In this process, the wafer is coated with a very thin layer of photosensitive chemical.  This layer is then exposed in a very intricate pattern that's projected off of a mask with short wavelength light.  The set of masks used determines the chip design, they are the ultimate product of the chip design process.  The feature size that can be projected onto the photoresist coating on the wafer is determined by the wavelength of the light used.  Once the photoresist is exposed, it is then developed to expose the underlying surface.  The exposed areas can be operated on by other processes - e.g. etching, ion implantation, etc.  If photolithography does not have enough resolution, then there is another technique that uses focussed electron beams to do the same thing.  The advantage is that no masks are required as the geometry is simply programmed into the machine, however it is much slower as the beam (or multiple beams) must trace out each individual feature. 

The transistors themselves are built up from several layers.  Most chips these days are CMOS, so I will briefly describe how to build a MOSFET transistor.  This method is called the 'self-aligned gate' method as the gate is laid down before the source and drain so that any misalignment in the gate will be compensated for.  The first step is to lay down the wells in which the transistors are placed.  The wells convert the silicon into the correct type for building the transistor (you need to build an N channel MOSFET on P type silicon, and a P channel MOSFET on N type silicon).  This is done by laying down a layer of photoresist and then using ion implantation to force ions into the wafer in the exposed areas.  Then the gate oxide is grown on top of the wafer.  On silicon chips, the oxide used is generally silicon dioxide - glass.  This is done by baking the chip in an oven with oxygen at high temperature.  Then a layer of polysilicon or metal is plated down on top of the oxide.  This layer will form the gate after it is etched.  Next, a photoresist layer is put down and exposed.  The exposed areas are etched away, leaving the transistor gates.  Next, another round of photolithography is used to mask off the regions for the transistor sources and drains.  Ion implantation is used to create the source and drain electrodes in the exposed areas.  The gate electrode itself acts as a mask for the transistor channel, ensuring that the source and drain are doped exactly to the edge of the gate electrode.  Then the wafer is baked so that the implanted ions work their way slightly under the gate electrode.  After this, the transistors are complete and the wiring layers are built up one after the other.  

I dug up a couple of decent videos that are in fact educational videos and not PR videos:

http://www.youtube.com/watch?v=35jWSQXku74

http://www.youtube.com/watch?v=z47Gv2cdFtA