So if an enemy does 1d2 Strength damage through poison and I the DM roll a 2, does that mean that the player has to subtract 2 from their Strength statistic, or do they just take 2 damage, or both? And if the player has to reduce the their Str, when do they get those 2 points back?
If something (a poison, a curse, a disease, etc) does ability damage to your character the damage is done only to the relative ability. So, in your case, 2 strength damage.
From the section of the Pathfinder SRD on Ability Score Damage:
Diseases, poisons, spells, and other abilities can all deal damage directly to your ability scores. This damage does not actually reduce an ability, but it does apply a penalty to the skills and statistics that are based on that ability.
For every 2 points of damage you take to a single ability, apply a –1 penalty to skills and statistics listed with the relevant ability. If the amount of ability damage you have taken equals or exceeds your ability score, you immediately fall unconscious until the damage is less than your ability score. The only exception to this is your Constitution score. If the damage to your Constitution is equal to or greater than your Constitution score, you die. Unless otherwise noted, damage to your ability scores is healed at the rate of 1 per day to each ability score that has been damaged. Ability damage can be healed through the use of spells, such as lesser restoration.
Strength Damage is a type of Ability Score Damage that creatures can take.
It is a different type of damage than regular, non-lethal, or other forms of Damage.
Each Ability Score damage should be tracked separately from each other and normal damage. Strength Damage has specific penalties every time it increases to an even number:
For every 2 points of damage... [a creature takes a] penalty on Strength-based skill checks, melee attack rolls, and weapon damage rolls (if they rely on Strength). The penalty also applies to your Combat Maneuver Bonus (if you are Small or larger) and your Combat Maneuver Defense.
Additionally, if you take Strength Damage equal to your Strength Score (not Modifier) you are rendered helpless:
A character with a Strength score of 0 is too weak to move in any way and is unconscious. Some creatures do not possess a Strength score and have no modifier at all to Strength-based skills or checks.