In the City of Magnimar there are a large number of factions, some working alongside each other, some working against each other. Making a name for yourself will probably involve currying favour with some of these factions in order to gain access to resources, help and fame.
There are a number of sets of rules for Reputation and Fame in Pathfinder, which seem to have changed over time. I'm using a slightly modified amalgamation of those in Pathfinder Chronicles: Faction Guide, and Ultimate Campaign, and applying them to each faction.
Note that a PC may begin play as a member of, or allied with, a particular Faction. This doesn't grant them any special position on the Reputation track for that Faction - they still begin with a Reputation of zero. They are untrusted newcomers who have yet to prove themselves. However, it may provide greater opportunity to begin proving yourself early on.
Who many of the Factions are is deliberately left undefined at this point, but some of them will be relatively obvious (the Pathfinder Society, Aspis Consortium and major organised religions in Magnimar are probably good candidates).
Information on the factions which has been revealed during plan can be found in these pages.
A character's standing with each faction is mechanically represented by their Reputation score. Whenever you perform a service for a faction (providing them with information, completing a mission, etc) then you will gain 1 or more points of Reputation.
Each point of Reputation will also give you one Prestige Point (PP). Prestige can be spent to gain benefits, but your Reputation stays as it is.
As your Reputation increases, so you will unlock new benefits that you can take advantage of. These may be a skill bonus when dealing with members of the faction, the option to take prestige classes, opportunities to obtain equipment or access to resources, or special one off benefits unique to that faction. Some of these will have a PP cost associated with them.
The benefits provided by each faction will not be known until they are unlocked by gaining sufficient Reputation.
There is no cap to how high a Reputation score can go, but all Factions will have a point where further benefits will not be gained. For a small faction, this may be 20-30, for a larger faction it may be 60-80.
If you take actions that work against a Faction, then you could lose Reputation. Your Prestige Points can never be higher than your current Reputation, so if the latter drops below your PP, then you will also lose PP.
Not all Factions get along, and some are in conflict with each other. Whilst you are relatively unknown, they won't pay too much attention to who else you are working with. However, as your Reputation increases with one Faction, other Factions may start taking a disliking to you and you will lose, or prevented from gaining, Reputation with those.
When you gain ability training, then you gain further points to put towards an ability score using the point buy method. The cost is whatever the cost would be to raise your ability score to the next level, to a maximum level of 18.
If the cost is greater than 1, then you must pay the full cost in one go. So to increase an ability from 12 to 13 normally costs 3 points, so if the PP cost is 10 (it may vary from faction to faction), you must be able to spend 30 PP all at once, and they must all come from the same faction.
Note that for below average abilities, the cost is the number of points you would have gained to get your current level. So it costs 4 points to go from 7 to 8, 2 points from 8 to 9, and 1 point from 9 to 10. If you have an ability score below 7 (due to a racial modifier), the cost is treated as 4 points per level.
There is no limit to the number of times you can raise an ability score in this way except for the cap of 18, and it does not affect inherent bonuses or raises due to level.
Many benefits grant circumstance bonuses to skill checks in certain situations. Circumstance bonuses stack with each other unless they come from the same source. All bonuses from a particular faction count as the same source, so they won't stack with each other.
Circumstance bonuses granted by different factions will stack with each other.
If you gain skill specialisation, then that skill immediately becomes a class skill for you. If it is already a class skill, or you later gain it as a class skill from another source, then you gain a +1 competence bonus in that skill.