IVs stand for Individual Values, and each stat a Pokemon has can possess an IV with a maximum of 31. If caught in the wild, the IVs are randomly set between 0 and 31, but if bred, the values can be inherited by their parents. IVs raise the stats of the Pokemon exactly according to its value, so a perfect IV Pokemon in Attack would have 31 more than one with 0 in Attack, as shown in the stat itself. To pass down IVs, one of the parents must hold a Destiny Knot, which can be obtained by the couple in Sea Mauville. Without this item, the parents can only pass down up to three of their IVs, the other three determined randomly as if the offspring has been caught in the wild, but with the Destiny Knot, five can be passed down. The IVs can be checked by the judge in the Battle Resort Pokemon Center. If a Pokemon is "outstanding," then it has very high IVs overall, and if the stats he list "can't be beat," then they are perfect. Any negative comments on a stat means it's 0. Below outstanding is relatively superior, above average, and decent, and below can't be beat is fantastic, definitely good, and decent.
Natures are also significant in competitive. To pass down a nature, if one parent holds an Everstone, which is easily obtained in the storyline, it passes down its nature to the offspring. Natures can lightly raise one stat by 10% and another by 10%. There are a total of 25 natures, and five are neutral natures: Bashful, Docile, Quirky, Serious, and Hardy. The HP stat cannot be affected by natures. As for the other five, there is one nature that decreases one stat and increases another.
What you need to breed a Pokemon is access to a Daycare, and two parents of the same species, one male and one female. The offspring will always be the first Pokemon of the mother's evolutionary line, with the exception of baby pokemon like Mantyke and Bonsly. Baby Pokemon require a certain item held by the mother. All pokemon are given to the player in the form of Eggs, and each pokemon hatches at different rates. All rates can be quickened if the leading Pokemon in the party has the ability Flame Body or Magma Armor. Pokemon hatched are always level 1. Some Pokemon can inherit Egg moves, which may not be obtained in its levelup moveset. They can be inherited from both parents, but the father's are usually superior. The female's ability affects the offspiring. If it has the hidden ability, there is a 60% chance the offspring will inherit it, and a 40% chance of a regular ability. If the mother has a regular ability and it has a second regular one, there is an 80% chance to pass down its ability and a 20% chance for the other one.
Pokemon without a gender that are still in a species can only be bred with Ditto, another genderless pokemon. Unlike others, Ditto can breed with both male and female Pokemon if not genderless, the offspring produced not being Ditto. Likewise, Ditto's nature and IVs can be passed down the same way. All pokemon in the Undiscovered group, mainly consisting of Legendaries and Baby Pokemon, cannot be bred at all.