I'm not a tech person...I'm just a user like you...so I'll give my view
Well...generally speaking...the key match between motherboard and CPU is the socket...so if you have a socket 775 processor, it's safe to assume that any socket 775 board will work.
But...after you assume that, it's easy to take a look at the specs for the board...and the CPU
Chipsets...matter, in some cases. The chipset is important because it will reflect your onboard functions...and some chipsets have idiosyncracies.
In your case, it's pretty simple.
Intel lays out all sorts of data re this board, http://www.intel.com/p/en_US/support/highlights/dsktpboards/d945gnt
, including Supported Processors.
By looking at either of these...you can find a fit in either CPU or board.
<<Also, i would like to have a general opinion on the two motherboard brands - Gigabyte or Asus, which one in your opinion is generally better?>>
Depends...on what board, what the user intends to do or wants to do...there is no honest blank endorsement that I can give for a brand. That's (IMO) one of the big fallacies that users make...they think that a board which is good...is typical of the quality of ALL products by that manufacturer...and the converse...that a board which has problems is typical of ALL boards by same manufacturer. It doesn't work that way, IMO.
I won't bore you with my personal preferences, but it would be neither of the manufacturers you listed. That doesn't mean that my preferences result in "better" boards used by me...probably the reverse
, I find that I like boards that others here scorn...I just treat motherboards for what they are...a product that I want and need and I don't need a lot of bells and whistles that others may want...for overclocking, game-playing, RAID, etc...because I'm not interested in those aspects of computing.
I do/have had boards by both Asus and Gigabyte. If I had to choose one or the other for a build/replacement...I would simply go with the cheapest which matched my needs and which happened to be available at the moment I decided to spend dollars
.Tiger Direct Socket 775 BoardsNewegg Socket 775 Boards
The above are not the only vendors, they happen to be two reliable vendors regarding computer parts, current and older.
This will get you started on your quest for answers