Pokémon have several hidden stats, let's just go through the ones we can see on the stats page.
HP (hit points) is related to how much damage a Pokemon can sustain before fainting.
CP (combat power) is not related to how much damage a Pokemon deals when attacking gyms, but is a combination of attack, defense, and stamina (HP). If a Pokémon would have less than 10 CP, it has 10 instead.
Height and weight, despite all the rumors that they affect gameplay, are purely cosmetic and have no gameplay impact (other than the Fisherman and Youngster medals).
The blue glow that appears behind Pokémon in your storage mean that you caught it in the last 24 hours.
Note that evolving a Pokémon retains the same relative value for HP and CP that increase by a species-specific ratio because their underlying hidden stats (IVs) don't change, whereas height and weight are randomized from scratch.
Pokémon have a hidden stat called a "level".
The white semicircle above them on their stats page shows their progress from minimum level (level 1 on the left side) to maximum level (player's current trainer level + 2, up to a maximum of level 40). Every time a Pokémon powers up they gain 1/2 a level and some amount of CP, and the white dot moves farther to the right end.
If it reaches the end (and the trainer is not yet at least level 38), it will say that the trainer needs to increase their trainer level to power up further (and thus allow the Pokémon to level up more, thereby gaining more CP). This is because a Pokémon's level is not allowed to exceed a trainer's level + 2. If a Pokemon is powered up to level 40, the Power Up option disappears.
Powering up costs both Stardust and candies specific to that Pokémon type (or by using Rare Candy that acquired from defeating Raid Bosses). When it is powered up, the Pokémon gains CP following a complicated formula, gaining more CP the better its hidden stats are.
Whenever a Pokémon is powered up, its level increases by 1/2. A Pokémon at the minimum possible CP is at level 1. To calculate a Pokémon's level, divide its current CP by how much CP it gains per power up, and then divide that again by 2.
Every 2 levels (or 4 power ups), the Stardust cost of powering up increases. Every 10 levels (or 20 power ups), the candy cost of powering up increases by 1. Note that since level determines Stardust cost, it is possible for a lower CP Pokémon to cost more to upgrade since it can gain less CP per level and thus have a lower max CP cap.
Don't forget, a Pokémon cannot be powered up past a trainer's level + 2. And even if a trainer achieves max trainer level, the only way to get the most powerful Pokémon is to find one with perfect hidden stats called Individual Values (IVs), which will be covered more in the advanced stats article.
Every Pokémon has a random Fast move and a random Charge move that is chosen from a list of valid moves. The Fast move is faster but usually deals less damage. The Charge move usually deals more damage but is usually slower and needs to be charged.
The width of the charge bars indicate how much charge a Charge move requires (more meaning more charge required), and the number of bars indicate how many charges can be stored at once.
Every move also has a hidden attack speed, and generally lower damage moves have faster attack speeds.
Using Fast moves and Charge moves that are the same type as the Pokémon gives a 1.2x damage multiplier called same type attack bonus (STAB).
Note that evolving a Pokémon randomizes its moveset.
Evolving costs a large number of candies, but no Stardust, and grants 1 candy.
When a Pokémon evolves, its base stats change so the displayed HP and CP increase. However, its Pokémon level and IVs do not change, so when a naturally powerful basic Pokémon is evolved, its evolution will also be naturally powerful.
In general, it is preferable to evolve a Pokémon before or instead of powering it up since evolving a Pokémon gives experience whereas powering it up does not. Also, a trainer can then choose to power up a Pokémon that has the desired moveset, rather than powering up a Pokémon only to find that its evolution has less desirable moves. (UPDATE: With the advent of gym raids, trainers have a chance to win Technical Machines [TMs] that allow you to change Fast or Charge moves, which takes away some of the stress of evolving and the randomized movesets that result from evolving.)
Pokémon have the following hidden core stats, similar to the games:
Each has a base value (predetermined by species) and an individual value (IV) that is added on top of the base value and ranges from 0–15. To have a truly maxed out Pokémon requires acquiring one with maxed out IVs.
For a more in-depth look at these stats, check out the breakdown here!