1. Around Feb each year, every club is asked to submit preferences for dates and matches (i.e. we can't be at home on Reading Festival saturday, Doncaster can't be at home on St Leger Day, there's a flower show in Shrewsbury which means they lose the car park so have to be away ..... and so on - there are hundreds of such requests!).
2. All police forces do the same, and any other requirements they have, like "pairings" (e.g. both Sheffield Clubs, Liverpool & Everton, the two Bristols, etc. can't be home on the same day. In London there are 4 or 5 way "pairings", and also police rules that "high profile" matches (e.g. London derbies can't be on bank holidays.) Also Blackpool, Brighton, etc have to be away when party conferences are on in their town, etc, and certain clubs aren't allowed to play at a seaside town on a bank holiday.
3. These requests are all allocated priorities, and, under the fixture compilation rules, govern the positions in the "fixtures grid" that each team must take. This basically defines the sequence of homes and aways for each team - although not all club and police requests can actually be met, about 87% are.
The compilation rules vary slightly each season, but an extract from these from a few years ago is :
1.2 Consecutive Home or Away Fixtures
No Club should play 3 consecutive home or 3 consecutive away fixtures.
1.3 Sequencing over Five Fixtures
In any sequence of 5 fixtures a Club must have either 2 home matches and 3 away matches or 3 home matches and 2 away matches.
1.4 FA Cup Rounds
Where possible, either side of Rounds One and Two of the FA Cup, Clubs in Leagues 1 & 2 should have a home and an away fixture. In addition, where possible, both FAPL and Championship Clubs should receive a home and an away fixture either side of Rounds Three and Four.
Where possible, a similar arrangement should apply to Premier League and Championship Fixtures either side of a blank Saturday for Internationals.
There's a whole lot more to these compilation rules than just that, though.
4. At this point the computer takes over, and allocates teams to positions in the grid until it finds combinations which fit.
5. The fixtures working group (which has representatives from the appropriate league and the FSF) then do a manual check of the fixtures and, where there's horrendous errors try to make changes, although this is usually really difficult, because changing one fixture will affect at least 7 others and usually more, because of the pairings. It's like pushing down a bubble in wallpaper - another bubble comes up somewhere.
That's it in a nutshell - having seen it all, I find it amazing with the number of police and club requests made that they do actually manage to find a set of fixtures that work.
There's certainly no favouritism for any club from FA, FAPL, FL or conference.