When a Ponymon reaches a certain requirement it has the possibility to undergo evolution which lets it evolve into a new Ponymon. All evolutions only improve existing stats. With soem exceptions most evolutions can always learn same moveset as their pre-evolved form could. But evolved ponymon learns new moves at later levels.
Evolved Ponymon also gain more base stats than their pre-evolved form.
Somtimes when a ponymon evolves it will also gain a new type. In some few cases evolved forms go from being pre evolved form types x+y to x+z.
The most common requirement for Ponymons which can evolve to do so is to reach a certain level. Other requirement are the use of stones or through gaining enough Happiness.
If you do not want your Ponymon for evolve for various purposes (Want to learn a move it can only have in pre evolved form, Think they look cuter, Trying a challenge etc etc) Then during the earlier parts of the evolution process you can press the B-button to cancel the evolution. When you cancel you only cancel for that level and during your next levelup your Ponymon will try to evolve again.