These are basically moves that are passed down from a parent to an offspring Pokemon.
However, there are only some specific moves that are passed down from the parents.
For Eg.- We have two Sesmitoads, a male and a female, and the male one has the moves Drain Punch and Earth Power. Breeding them together will result in a Tympole which knows Earth Power but not Drain Punch. This is because Tymople does not get the move Drain Punch as an egg move. See more here.
These are usually moves which Pokemon cannot get through normal means, such as the Earth Power Tympole as stated in example. In Gen 5, only males can pass down egg moves, but in Gen 6, if breeding with a Pokemon of the same species or Ditto, females can pass down egg moves as well.
IVs (Individual Values):
These are genetic factors in Pokemon, to be short. These factors determine how much will the Pokemon have in it's stats. For Eg.- a Pokemon with high IVs in it's Attack stat will have a comparatively higher Attack as compared to normal. The highest these factors can go is 31. See more information here.
When two parents are bred, a random pick of three of their combined IVs are taken and given to the offspring. Hence, breeding IVs into a Pokemon is quite a bit of the time, based on chance. However, there are a few techniques to breed in IVs into a Pokemon:
>1. Giving one of the parents Destiny Knot. This item increases the IVs that are tranferred on the offspring, from a random pick of 3 to a random pick of 5.
2. Having a parent hold a Power Item for respective stats. Giving a parent a Power item ensures that the IVs corresponding to the Power Item are passed onto the offspring. A parent holding the Power Anklet will ensure it's speed IVs will pass on to the offspring.
Natures stand for personalities of Pokemon. They simultaneously increase a stat and decrease another one. For Eg.- An Adamant nature increases the Attack stat of a Pokemon more than normal, while reducing it's Special Attack to become below normal. There's a simple way to tranfer a nature onto an offspring, that is Everstone.
Before Gen 6, when Everstone is held by a parent, there is a 50% chance that the offspring will have the same nature. In Gen 6, this has increased to a full 100% chance, which means that the nature will be passed on.
This method was introduced by the maker of this game, Junichi Masuda, to promote Pokemon trading across continents/countries. This method is popularly used to hatch a Shiny offspring. The method is this:
> When a two Pokemon from different regions breed, there is a higher chance of the offspring being alternate coloured.
So if a Japanese Pokemon breeds with a German Koffing, there is a higher chance that the Koffing that hatches out of the egg will be shiny. However, this method increases the rate of shiny encounter by 5 (from 1/8192 to 5/8192 or approximately 1/1638) which means that it is still based on luck. See more information here.
These are special items that help in breeding some particular Pokemon. For Eg.- A female Snorlax without the Full Incense, no matter how many times bred, will never result in a Munchlax, but rather another Snorlax. However, if the Snorlax is given a Full Incense to hold, the resulting Pokemon will always be Munchlax.
>This is also a way to breed in some egg moves onto some Pokemon, since the different offsprings can have different egg moves. For Eg.- Marill cannot learn Fake Tears through breeding, however, Azurill, the alternate option for an offspring, can. In the same way, Marill can learn Aqua Jet through egg moves while Azurill cannot.
More information over here.
Hope I helped!