Re: how does a light bulb work and who invented it?

The first problem here is what do you mean by "light bulb"?  I am going to
presume you are asking about the incandescent light bulb.  That is a light
bulb with what is essentially a wire inside that gets hot and emits light
(as opposed to say fluorescent, or LED or halogen light bulbs).

Light bulbs work by passing a current through a filament.  The filament
gets very hot, hot enough to emit visible light.  The filament is kept
inside a bulb filled with an inert gas.  This prevents the filament from
oxidising (burning).
 http://www.madsci.org/posts/archives/1997-05/864507907.Ph.r.html

Firstly lets dispel a bit of a myth.  Thomas Edison did not invent the
light bulb.  He however did adapt it, much improving it, to make what was
to become the first commercially practical incandescent light bulb.

Most historians attribute the invention of the first electric light to
Humphry Davy and English scientist and inventor in 1809.  This was an "arc
lamp" though, different to what most people think of as a light bulb.  Arc
lamps are still used today.  The light is produced from an arc of
electricity, not from a hot filament.
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 http://inventors.about.com/od/dstartinventors/a/Humphry_Davy.htm

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Davy also produced "incandescent" light by passing a current through a
platinum coil.

Not long after Davy, scientists and inventors began experimenting with
incandescence.  Producing light from a wire made hot by passing a current
through it.

The problem with these early incandescent lights was the amount of light
produced, the melting point of the filament, the combustion of the filament
(stopping them from burning), the means of mass production, and finally an
electrical system for the lights to run from.

In 1820 Warren De la Rue enclosed a platinum filament inside an evacuated
glass tube.  It worked better than most other efforts but platinum is
expensive and longevity was still an issue.

Around 1835 James Bowman Lindsay also demonstrated an incandescent bulb. 
He did little other work on this though.

Over the next 30 years a number of others demonstrated incandescent bulbs.

Around 1875 Joseph Swan started producing incandescent bulbs with a
significant life span (over 13 hours).

Edison then greatly improved the light bulb.  Much of it with the help of
Lewis Howard Latimer.  Both men taking out patents on their inventions. 
Around this time large electrical distribution networks started being
produced.  This greatly enhanced the commercialisation of the light bulb.
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 http://inventors.about.com/library/inventors/bllight2.htm
 http://www.unmuseum.org/lightbulb.htm

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So it is hard  to say who was the very first.  It in a way depends on
exactly what you mean by "light bulb".  Like many inventions it was a
series of small steps, each one working off the results of others before them.