What makes one microphone better than another, and how do I make my own?
Asking what makes one microphone better than another is like asking what makes one car better than another. This is a HUGE subject that I can only give an overview for.  But here goes:

Diaphragm

Of course the diaphragm is the most important (but just barely).  This is the part that moves in response to the sound and somehow converts the sound to electrical signals.  This is the part that determines if your mic is a dynamic, condenser, ribbon, or other type of mic.  

To get good frequency response the diaphragm must be very light.  The lighter the better.  But the lighter something is, the more fragile it is.  So steps have to be taken to make it more robust and/or to protect it from damage.  

A good diaphragm will also resist environmental damage.  Mostly humidity, but also smoke, temperature extremes, shock, etc.

Building a diaphragm is super difficult and requires specialized materials and machinery.  This is beyond almost every hobbyist, and even most companies.  Odds are that more than 75% of the mic brands sold in places like Guitar Center do not make their own diaphragms-- and it could be as high as 90%. 

It is possible to make a diaphragm in your home, but it won't sound good at all.  You would be lucky to make one that has "telephone quality".

Sometimes the diaphragm is called the "mic capsule", but that can sometimes be misleading.

Mechanical

The mechanical properties of the mic should not be underestimated.  By mechanical stuff, I mean everything that is not the diaphragm or electrical.  The body of the mic, the grill, the positioning of the diaphragm inside of the mic and even the mic clip.

It is primarily the mechanical design that determines if the mic is a cardioid, omnidirectional, or hyper-cardioid.  Probably more to the point, it is the mechanical design that determines how much unintended noise is rejected.  This is done through the placement of the diaphragm and the design of the grill.

Another important task of the mechanical design is to reduce handling noise-- noise caused by someone holding, moving, or adjusting the microphone.  In really bad mics, just flexing your hand while holding the mic will be clearly heard.

The mechanical design also helps protect the mic from wind, breath, spit, smoke, beer, etc.

Electrical

Most mics, particularly condenser mics, have electronics in them to boost the signal from the diaphragm before outputting it.  Some mics also have simple switchable high-pass filters that further reduce handling and wind related noise.

Of course the electronics have to be low-noise while being able to handle a huge dynamic range without distortion.  Even though these circuits are usually very simple, making them high quality is easier said than done.

This gets even harder for the USB-Mics that are coming out.  Making a consumer grade USB mic isn't too hard, but a studio grade USB mic is difficult due to the extra electronics.

Consistency

Doing the above things consistently is super important for a microphone manufacturer.  Once you have the expensive manufacturing equipment it is relatively easy to make a quality mic.  But making two quality mics that match each other in output level, frequency response, transient response, etc. is difficult.

This is super important if you need a pair of mics for stereo recordings.  If the left mic has a slightly different frequency response than the right mic then the recording will sound weird.  Almost every mic company will sell "matched pairs" of mics, where they go through a batch of mics and find two that are very closely matched.  Other companies won't even bother since they have a hard time making 2 that are the same.

Building a high quality mic at home

This is super difficult, but depends on how much you want to do yourself.  Making one from scratch is almost impossible because it requires skills and equipment that is rarely found in one person (vapor deposition, electronics, metal fabrication, plastic and/or rubber injection molding, etc.).  

You could take one one of those tasks and rebuild an existing mic with that task.  For example, you could take a cheap Chinese mic and rebuild or redesign the electronics in it.  Or get a metal lathe and milling machine and redesign the body of an existing mic.

It is certainly possible, with some careful research and planning, to take a US$75 mic and make it perform like a US$500 mic.

But don't bother trying to make your own diaphragms if audio quality is your goal.  By all means make one just to learn stuff, but don't expect it to sound good.

El-Cheapo Condenser Mic Capsules

These are not useful for much when it comes to professional audio.  The quality just isn't there, in noise, frequency response, dynamic range, etc.  Professional condensers have a 0.5 to 1.0 inch diameter diaphragm in order to be sensitive and have a better frequency response.  The cheap ones are usually less than 0.25 inches, and possibly 0.10 inches.

Mic Arrays

One emerging area of mic research today is the use of microphone arrays.  This is where many el-cheapo mics are used and combined in ways that are useful.  Normally the mic arrays are used to make a super-directional microphone for teleconferencing and similar applications.  This is an area that someone at home can more easily play in and possibly come up with something that works better than what is available commercially.