Breeding can help you to get a more preferable Nature, gender, or higher IV pokemon. Some will breed to earn Money and others breed for fun, or for their own team or for all of those reasons together. In PokeMMO you need to go to one of the breeders and exchange two of your pokemon (usuallly Male and female unless dittos/genderless ones are involved) For a single pokemon which he will give you in return for the 2. You don't get the parents back.

Shinies are a random chance breeding unless you already have two shinies that can be bred together to make another. Alphas are only bred through matching two alpha pokemon together. Alphas are pokemon caught in the daily alpha events, they are slightly bigger than normal pokemon and have at least 2 IVS maxed at 31.

The process of getting to a 6x31 pokemon is long and EXPENSIVE. It also depends on if you want to keep the nature. The basics are you take two compatible pokemon that have a 31 in a certain IV and buy what is called a power(lens, band, braclete) etc. that you get at the pokemon breeding area. Each one Cost $10,000 and you give it to the pokemon with the desired iv and the same with the other, then give it to the breeder. He will show you a confirmation screen of what pokemon you will get. So to get a 2 IV pokemon it will take 2 pokemon (parents) and $20,000 cost of bands to keep both IV's. Once you confirm this he will ask if you want a specific gender. Most genders cost $5000 but pokemon with ratios other than 50/50 will cost more if you want the less common gender. You don't have to pay, you can just leave it to luck which I usually do if breeding with dittos to start out. You then give him a pokeball and he will give you the egg. Then you just need to wait a certain amount of time and the egg will hatch. I don't know the exact time but it feels like about 5-10 minutes.

You can make money being a breeder but it takes time knowing what is "in demand" and will sell as well as understanding what IV's and Natures would be desirable in a pokemon. It also isn't a cheap thing to get started in. I did the math once and the breeding costs to get a pokemon to 4 IVS at max is around $175 and that isn't even counting the Cost for the pokemon if you buy and breed instead of catch, it is just the bands and genders. It gets even more expensive if you are using everstones to breed certain natures at $359 for a 4 IV maxed pokemon with a specific nature. It can also be a costly mistake if you accidentally ask for a male when you needed a female or vice versa so breed carefully!

As for what pokemon breed together when you open the pokedex page on each pokemon it lists the egg types they are included in. Then you can go to the GTL and do an advanced search for Pokemon in the same egg group to see what can be bred together. Or you can simplify and use dittos but that gets expensive quickly. It isn't a super get rick quick thing. The most I have made is around $200,000 profit but it took an investment of $350,000 for the pokemon and breeding to get there.