In Pokemon Go, your hard caught Pokemon always have an opportunity to become a better form of themselves. Each Pokemon has a different CP, or combat points, which is a basic way to determine their usefulness in battle against other Pokemon. The higher your CP, the more likely it is your Pokemon will put up a valiant fight.
When you first catch a Pokemon, you will notice their CP rating. If a friend nearby catches the same Pokemon in their game, it is very likely the CP will be different. For example, a Horsea you caught showed 50CP, but the same Horsea for your friend may be 23 CP. The starting CP depends on what level your trainer is, but the power up will still remain the same between the two of you.
If you have enough specific candies and stardust for your Pokemon, you can select power up in the Pokemon’s menu. Both can be acquired via catching Pokemon, completing research tasks, and other activities in the game. Powering up is essentially the equivalent to leveling up a Pokemon in the mainline games, and will raise the CP of the selected monster while also give it a slight health increase. Your Pokemon will have a limit to its CP, depending on what Pokemon it is and how far in the evolution chain it is. Once you reach trainer level 35, the CP limit will be removed and Pokemon can be powered up as much as you want, provided you have the stardust and candy needed.
Every Pokemon you have will have its own power up level that will determine the amount of CP gained per power up. This hidden number is determined depending on the stats of your Pokemon you see when you appraise it, and the catch rate of that species. Each power up increases the power up level by 0.5, up to the player’s Trainer level plus two.
After every fourth power up you do on a specific Pokemon, the needed stardust will increase. Once the needed stardust reaches 5,000, the increase in CP and HP will decrease, making power ups not as significant. As your Pokemon reaches higher CP levels, the candies needed to power up will also slowly increase, though not quite as often as stardust increases. If you want a complete table showing how much stardust and candy will be needed per power up, check out the table below. To keep things simple, the more you power up a Pokemon, the more stardust and candies will be needed, and the less CP and HP you will gain from the power up.