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 you can sustain before dying.
CP (combat power) is not related to how much damage you deal 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.
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 (your current trainer level). 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, it will say that you need to increase your trainer level to power up further (and thus allow your Pokémon to level up more, thereby gaining more CP). This is because a Pokémon's level is not allowed to exceed your trainer level + 1.5.
Powering up costs both Stardust and candies specific to that Pokémon type. When you power up, the Pokémon gains CP following a complicated formula, gaining more CP the better its hidden stats are.
Whenever you power up a Pokémon, their 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 you gain 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, you cannot power up a Pokémon past your trainer level + 1.5. And even if you're max trainer level though, 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 quick move and a random charge move that is chosen from a list of valid moves. The quick 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 quick 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 if you had a naturally powerful basic Pokémon, 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, you can choose to power up a Pokémon that has the moveset that you desire, rather than powering up a Pokémon only to find that its evolution has moves you didn't want.
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 you need to acquire one with maxed out IVs.
For a more in-depth look at these stats, check out the breakdown here!