The VDex daycare works mostly the same as the Pokemon games, but has a few very different and unique niches that make it far more user-friendly and useful. The basics of the daycare are simple: Leave two Pokemon of opposite gender in the daycare, and if they are of the same Egg Group, they will produce an egg over time.
A VDex exclusive item called the Egg Scope will allow you to see the species, gender, nature and sprite of the Pokemon that the egg will contain before you take the egg, as well as allow you to discard eggs (for a fee), which allows for specific breeding.
A shiny Pokemon holding Brightpowder has a 5% chance of passing it's shininess to the egg Pokemon, which is very useful in breeding shiny Pokemon. Shiningpowder works the same way, but has a 10% chance of passing the shininess down.
A Pokemon holding an Everstone has a chance to pass down their nature and sprite type to the egg. The percentage ratio depends on the sprite type attempting to be passed down.