In Pokémon Emerald, if the mother (in a male-female pair) or Ditto (in a pair that includes Ditto) is holding an Everstone, the baby has a 50% chance of inheriting that Pokémon's Nature.
From HG/SS and onawrd you will always have that chance to pass down the nature, regardless of its gender and whether it's breeding with a Ditto or not.
Starting in Black and White 2 you can give one of the two Pokemon breeding an Everstone and it will pass it down 100% of the time.
Normal abilities are passed down all the time, but which ability it gets is randomized.
From Black and White 2 onward, there is an 80% chance that the offspring will have the female's ability.
As for hidden abilities, from generation 5 onward there is a 60% chance that if the female Pokemon has the hidden ability it will be passed down.
In Generation VI, any Pokémon, including male and genderless Pokémon, can pass down their Hidden Ability when bred with Ditto.
A Pokemons IVs are always passed down, but how many and which ones it passes down changes between generations.
In Gen 2 games, the offspring will inherit the IVs from the parent that is the opposite gender from it.
If a male or female bred with a Ditto, it will inherit the Ditto's IVs.
The Defense IV is passed, and also either the Special IV or the Special IV plus or minus 8 (plus for values in the 0-7 range; minus for values in the 8-15 range); the Special IV has a 50% chance of remaining the same.
Attack and Speed IVs are determined entirely at random, whereas HP is determined by the IVs of the other four stats, using the same formula for wild Pokémon.
In Pokémon Ruby, Sapphire, FireRed and LeafGreen, three of the IV stats will be passed down from both parents together.
In Emerald it is a bit more complicated:
In Pokémon Emerald, the process is similar but more complicated. First, a random IV is passed from one of the parents to the baby. Afterwards, a random IV of any stat (other than HP) is passed from either parent to the baby; if it is the same stat (not necessarily the same IV value, depending on whether or not the same parent passed it on), it will override it. Finally, another random IV of any stat other than HP and Defense is passed from one of the parents to the baby; this IV will override any of the previous two if it is from the same stat. The remaining stats (a number ranging from 3 to 5) are determined at random, again with the possibility of one or more IVs coinciding with those of the parents.
Generation IV and V
In Diamond and Pearl it is the same as in Emerald.
In Platinum, the offspring will inherit 3 IVs from the parents. In addition to that, if one of the parents is holding an EV enhancing item(except the Macho brace), they will instead inherit that IV and then inherit two other random stats from each parent.
In generation 6, if one of the parents is holding a Destiny knot, the offspring will always inherit 5 random IV stats from both the parents together.
This can be combined with an EV-enhancing item to ensure that one desired IV is always inherited; however, the offspring will still inherit only a total of five IVs from its parents (one from the EV-enhancing item and four from the Destiny Knot).
Hope I helped!