Im planning to buy a 7670 Radeon HD graphics card.. but I came in to a dilemma.

What does a 2gb dedicated graphics memory do to the computer performance? as compared to... say.. 1gb dedicated graphics memory? 

a low-mid range card (like the 7670) won't profit from 2GB of ram and make sure the card you buy has GDDR5 ram - 1GB of GDDR5 is way better than 2GB of DDR3

and the term "dedicated memory" just means it's ram on the card opposite to "shared memory" where the card uses the system memory (which is slower and at most DDR3 aswell)

You really shouldn't go under radeon 7750 if you plan on doing any kind of current gaming.

What does it do?

Simply put the memory on a graphics card is a high-bandwidth cache that stores information such as textures and other data.
If it gets full, then that data will be stored in System Ram instead then streamed.

2gb of memory or more is most important when you are running extremely high resolutions like 5760x1080 or 2560x1600 with high-resolution textures.
Both of which a Radeon 7670 (Which really is a re-named Radeon 6670) is not able to handle.

Also worthy of note is that with lower-end cards you want faster memory not larger, lower-end cards with lots of memory usually go with GDDR3 memory instead of GDDR5 which has a massive impact on performance as the bandwidth is essentially cut in half.
If you can  find one, a Radeon 5770/6770 can be found really cheap which is a massive upgrade over a 6670/7670, otherwise go for a Radeon 7750/7770 if you can at a minimum.

It's not really a relevant spec. Every card you can find >$100 (and I wouldn't recommend going under that for gaming) will have enough memory to store the game textures.

Here's a chart of graphics card relative performance: http://www.tomshardware.com/reviews/gaming-graphics-card-review,3107-7.html

The rest of that article has some reasonable recommendations for price and desired resolution/settings.