Also @Person Man: there is NO REASON whatsoever to waste time powering up an unevolved Pokemon. Save your candies and go for the evolve.
Really? I thought there was a thing about how a pokemon's pre-evolution CP determined its max evolved CP. I could be grossly misinformed, though.
Yeah you are
HOWEVER it's pretty simple to explain, although it might ruin the "magic" of Pokemon. I spoilered it if you want to open Pandora's Box.
Spoiler, click to toggle visibilty
You sick fuck
If you're familiar with how the main series works, it's a dumbed-down version of that. Every species of Pokemon has Base Stats that never change. In Pokemon Go, your stats are Attack (how much damage you deal), Defense (how much damage you can sponge), and Stamina (how quickly you can fill your special attack meter, mislabeled as HP in appraisals). Despite this, it is very unlikely for two Pokemon to have exactly the same stats. When Pokemon are generated on the map or hatched, they get random stat boosts to each stat, ranging from 0-15, called Individual Values (IVs).
The CP value you see is calculated based on IVs and HP in relation to its level, which is randomly generated for the player based on their own level and other random factors. So for example, if you and your friend are walking together and find a Dragonair next to a water fountain or something, and yours is like 43 CP and his is like 1347 CP (I dunno CP ratios, these are just examples), they have the exact same IVs, it's just that his is at a higher level. The minimum level a Pokemon can be is 1, and the maximum is equal to whatever your level is (so the highest level a Pokemon can possibly be is 40). Additionally, HP is not its own unique stat. It's actually calculated by looking at the Pokemon's Stamina and multiplying it with their level.
Enter the Appraisal system. Based on what your team leader says, it will reveal how high the combined stat boosts are, which stat(s) have the highest value, how high that value is, and then... how light/heavy your Pokemon is (so far there's nothing discovered about weight/size's effects on stats). There's also online calculators I linked to in this thread that you can supplement this info with and get exact numbers.
NOW GRANTED, the biggest difference between a fully maxed out Pokemon and a Pokemon with 0's across the board is only 10%, but it's like Q's taunts in Street Fighter 3: It makes the difference between getting knocked out and having a sliver of health left.
That's only half of it though. ADDITIONALLY, your movesets are very important. Unlike the main games, with few exceptions (ie: Gyarados w/Dragon Breath or Blastoise w/Flash Cannon) you generally don't want moves that aren't your Pokemon's type because you don't get the Same Type Attack Bonus (STAB) from them, which is 1.25x. Type weaknesses/resistances are still in play though although their effects are diminished (something like x0.8 for resistance, 1.2x for weakness. Thank god type immunity isn't in this game). So ideally, you want a moveset that both capitalizes on your types and also deals a lot of damage very quickly. So for example, the ideal moveset for Jolteon isn't Thundershock and Thunder, but Thundershock and Thunderbolt because you can fire off Thunderbolts more frequently and recover faster than using Thunder.
tl;dr: A Pokemon's IVs and movesets determine their strength. You only want to pump candy into a fully-evolved Pokemon you're levelling up