Liggliluff explained it pretty nicely already, but I'd like to condense it a bit.

Whenever you have a definite noun (which is done in English by using "the", "this", "that", "my", "your", etc.), you need to use the definite article a/az in Hungarian. (The technical reason for that is that Hungarian does not have demonstrative or possessive adjectives, so you have to use some other method.)

• az autó - the car
• ez az autó - this car
• az az autó - that car
• az autóm - my car
• az autód - your car, etc.

"This is the beautiful car" would be translated as "Ez a szép autó". Notice the position of the adjective relative to the noun.

I can make a comparation to Swedish, where the adjectives changes depending on if it's definite or not.
Indefinite: vacker Definite: vackra

beautiful car – vacker bil
the beautiful car – vackra bilen
this beautiful car – denna vackra bil

Quick conclusion would be that this/that is definite, and you drop the "the".

The same for my:
my beautiful car – min vackra bil

So I wild guess is that you don't drop it in Hungarian, so you have "ez az autó", and "az autóm".

Not for this sentence, no. The Hungarian for your sentence is "Ez egy szép autó.".

The position of the adjective is important. Like in English, attributive adjectives are placed in front of the noun:

• a szép autó - the pretty car
• Az autó szép - The car is pretty.

"This is a pretty car" would be translated as "Ez egy szép autó".

Or do you mean the difference in meaning? The original sentence describes the car, and your sentence defines what "this" is.