In Pokemon Contest houses, you may enter your Pokemon into a contest for a chance to win a medal. However if you enter without any Porok level ups, you will have a very low chance of winning. The way to raise this is to go out and find the trees with berries on them, then press A. You will then receive 1 to 4 berries. If you now go to a Pokemon House, you will see small gray machines on the right. The empty one on the top is for multi-player. For now, find a house in a city that only has a man at the bottom. Go to the other side of the machine and press A. You will then be asked to pick a Porok (berry) and throw it in. The spinner will come down and start moving. The objective is to press A just when the spinner hits your arrow:

Circle Within A Circle: Direct hit (much faster)
Circle: Near hit (slightly faster)
X: Miss (slower)
The faster you (and your opponent) get the spinner moving, the better. Once the meter at the top hits the end, it is over and the results and the fastest RPM the spinner was moving will appear. You will then receive a Blended Porok. The level of it depends on how fast you got the machine going. The average appears to be 11. Next, open your Special Items Pocket and go to your "Pez dispenser" type item (which you must have to play the game -- it is obtained from the girl in the first contest house). You will then pick a Blended Porok and feed it to a pet, making one of his Pokemon Contest stats (viewable in the Poke-Navi) go up. Some can even raise multiple stats. The more you do this, the better chance you have of winning.