A pokemon's performance in battle is influenced by...

Mainly it's Combat Power (CP), level, skills and Individual Value (correlated to each other) (IV).

Combat Power

Every pokemon has 3 value to measure it's power: attack, defence and stamina. CP is the simplest way to measure a pokemon by its individual value, and it's easily understood with an algorithm:
CP = MAX(10, FLOOR(Stamina^0.5 * Attack * Def^0.5 / 10))
Someone may be curious about how Niantic design the IV for each pokemon, and it is proved that these 3 values are combined from the 6 individual values of the pokemon in generaion 6 by another algorithm. You can see the Reddit link for more details. Actually, though a high CP does mean that the pokemon is stronger, it cannot provide players with a complete comprehension to the pokemon, and we also needs the following factors:

Level (Correlated to CP)

You cannot see your pokemons' levels in the game, but we can guess or measure them base on several things:
1.a rule that pokemon's level must <= trainer's level + 1.5
2.Each time you power up, the level of pokemon will increase by 0.5.
Then we can guess the level of pokemon base on this sheet (the number of candies needed might be slightly different from actual):
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A high level of a pokemon means that its CP will also be comparatively high, and the stardust and candies you need to power it up each time will also keep increasing. After level 30, the benefit of power up will be reduced to half of before (like from 40CP/power up to 20CP/power up), and the cost will be quite high, which can be regarded as one of the penalty of a too high level.

Skills (Important and unrelated to other attributes)

Each pokemon has two kinds of skills: quick move and main move. And for most pokemons these two moves are randomly chosen between several possibilities. The difference of skills may influence the damage of pokemon by 10%-20% or even more; thus, skills are quite important for players who want to train a pokemon to be a core. CP can be increased by leveling up (power up), but skills cannot be changed by anyway when they are confirmed at the time you catch or evolve the pokemon. About if a quick move/main move is a good skill for a pokemon, you can view this website for the possible and best moveset combination for pokemons: link of the website
This website also has many other practical functions, which would be talked about later. For now, when you open the page of each pokemon in this website, you can see the quick moves and main moves it own, and their
DPS: The total damage (after considering critical strike chance and 25% same type attack bonus)/the time this skill needs
However, in a gym battle even DPS cannot precisely determine a pokemon's damage, especally considering that you may need to dodge the enemy's skill. Thus never be too serious about these numbers, they mean something but not everything.

Fluctuated Individual Value

This is not known by most players, even after the update of appraisal system (you can click the appraisal button in the right bottom of a pokemon's panel)
Besides the fixed attack, defence and stamina values, each pokemon has three 0-15 fluctuated values for these three attributes. Example: Let's say a Lapras has attack be 200, defence be 210 and stamina be 220 when it is level 40. If it's final IV is just 200, 210, 220, we say it's IV is 0%; if it's final IV is 215, 225, 235, we say it's a 100% IV Lapras. In short, the fluctuated IV, or IV%, is just the sum of 3 fluctuated values divided by 45 (15*3).
Undoubtedly IV% unchangeably matters the power of a pokemon, which cause many players to mindlessly chase for a extremely high IV% one. But after research, we find that a 90% IV pokemon is just around 2% less powerful than a 100% IV one, and 80% IV is 4% less powerful, and 50% IV is 9% less powerful. Therefore IV% is so not as important as skills, or we can say you don't need to worry it at all as if the pokemon doesn't have a quite low IV%. Another explicit example: A 100% IV Gyarados with Twister main move is worse than a 0% IV Gyarados with Hydro Pump main move. What's more, the website mentioned before also provides a function to measure(atually list all the possibilities) the IV% of a pokemon, and this function is much improved after it combined the official appraisal system and could usually located the possible IV% to 2 or 1 possibilities. The function is in the 'IV calculator' page in the website.
About appraisal system, I think it's quite a practical function, since it's highest rank is over 80% IV, and for most pokemons it is already unnecessary to check the specific IV% in websites if the IV% is known to be over 80%. You can check the following graph for how to use the appraisal system:

Type Weaknesses and Resistant

Will be mentioned in the skills in battle

Pokemon Go Guide

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