To raise Contest Stats, give a Poke'mon a poffin of that Stat, for example: I want my Empoleon to have more Beauty, so I give it a Dry Poffin.
Dry=Beauty, Sour=Smart, Spicy=Cool, Sweet=Cute, and Bitter=Tough.
Some Poffins are called something like Dry/Sweet Poffin. These will raise 2 Contest Stats.
The Mild Poffin raises all of them.
Raising Contest Stats makes the Poke'mon get more points in the first phase of the Contest.
If you give a Poke'mon a Poffin it likes, it's Happiness goes up. If it doesn't like it, the Happiness goes down. You can see what flavor the Poke'mon likes in the second page of the Summary.
To find out what your Poke'mon's happiness is, go to Twinleaf Town and talk to Nolan's sister.
I think you have to beat the Elite Four first, though.
I don't really know how to explain Contests, but I'll try my best.
General info on Contests.
First phase. Dress up.
In this phase, dress up you Poke'mon according to the theme. Example: The Natural is the theme. One good thing to dress up your Pokemon in would be the Dewdrop.
The Colorful is the theme. A good thing to dress up your Poke'mon with would be a Pink Fluff.
Make sure to dress up you Poke'mon with lots of things, but remember there is a limit.
Second phase: Dancing
In this phase, the lead Poke'mon dances, and when it s the're turn, the backup Poke'mon copy the lead Poke'mon's moves.
Example: Being the lead Poke'mon.
You can make your Poke'mon move by tapping the big circles at the bottom of the screen. In low contests such as normal, there are only 3. In the higher ones there are 4.
There are lines near the bottom of the screen. When your Poke'mon moves, it makes a dot on that those lines. The backup Poke'mon must put a dot in that exact spot to copy you perfectly.
To trick them, put the dots in between the lines(which is in between the beats). It is hard for the backups to follow.
Example: being the backup Poke'mon.
When the lead Poke'mon moves, it creates a dot(see above). You, as one of the lead Poke'mon, must watch where the lead Poke'mon put it. As a verticle line crosses a dot, tap the big circle at the bottom of the screen that the lead Poke'mon had. For example, if the lead Poke'mon jumped, tap the top circle at the right time, which is the Jump button.
Third phase: Moves
Impress the judges by using your Poke'mon's moves one at a time. You must choose a judge to perform to every turn. The less Poke'mon you share a judge with, the higher score you get. If all Poke'mon share the same judge, no points are awarded.
Choose the correct move or the correct time.
Example: If I am performing last, and it is the Beauty Contest, Whirlpool would be a good move for me to use, because it is a Beauty move, and, in Contests, gives me four points if I am last.