(We decided to move this guide from this thread.)
Effort Values (EVs)
Effort Values, or EVs for short, are points that increase each of your Pokemon's stats, like Attack, Defence, HP, etc. Along with base stats, natures, IVs and levels, EVs are used to calculate the stat totals you see in your Pokemon's summary screen. (Here is the formula that is used to achieve that.) In short, having more EVs makes your Pokemon more powerful!
Each Pokemon can have a maximum of 252 EVs in each stat, and a maximum of 510 EVs in total. The amount of EVs a Pokemon needs to gain one real stat point gains depends on its level — for the mathematically inclined, the exact amount is equal to
floor(400/level)**. Otherwise, know that at level 100, four EVs will contribute one stat point to your Pokemon's total, and this scales down linearly with level; so eight EVs equals one stat point at level 50.
Due to the above mechanics, a level 100 Pokemon can get up to 63 extra points a given stat, and up to 127 in total, from EVs. (You can find this by calculating
510/4 respectively.) Your Pokemon gain EVs mainly by defeating other Pokemon in battle; each species has an EV yield that dictates how many EVs it will hand out when it is KO'd, and in which stats. For example, defeating Pikachu will yield 2 Speed EVs for all Pokemon that gain experience from the battle (including level 100 Pokemon as of Gen 5). You can earn extra EVs for defeating Pokemon using Power items, Macho Brace and Pokerus, and you can boost EVs in a particular stat using Vitamins and Feathers. The main way to decrease EVs is by feeding your Pokemon EV-reducing berries.
With careful planning, you can control how many EVs your Pokemon gets, and in which stats. You can use this to strategically allocate EVs, creating the best advantage for your Pokemon. This is a key component of moveset building in competitive play. The common 252 / 252 / 4 EV spread is created when people choose to fully max EVs in two stats, and leave the last four meaningful points in another stat. This article on Smogon introduces the strategy of EVs in competitive play.
EVs have evolved over the generations. In Gen 1 and 2, EVs use a totally different and more primitive system that has no potential for strategy. Until Gen 6, the maximum number of EVs per stat was 255 (though this has no functional difference because Pokemon uses integer division). In Gen 7 onwards, it is possible to view a chart of your Pokemon's EVs by pressing a button in its summary screen. Also, the timing and frequency of stat recalculation (and thus when new EVs are 'applied') varies by game. Stats are always recalculated upon level up, but in more recent games, any change in EVs leads to an immediate recalculation of your Pokemon's stats.
** Subject to rounding in the stat calculation formulae, especially at low levels.
Individual Values (IVs)
Individual Values, or IVs for short, are semi-randomly generated numbers that are assigned to your Pokemon in each stat. Like EVs, IVs contribute to your Pokemon's stat total, but they are much harder to manipulate, especially in older games. You may think of IVs as your Pokemon's genes.
Each of your Pokemon's stats has an IV ranging from 0 to 31. Your Pokemon's IVs in all stats are generated using pseudo-RNG when you first encounter them — for Eggs, this means when you first pick them up. Like EVs, the effect of IVs on your Pokemon's stats scales with its level; at level 100, one IV is worth one real stat point. This means that at level 100, IVs can add as many as 31 points to each stat, and 186 in total.
IVs are a bother to competitive players. In most cases, it is ideal to have Pokemon with six stats fully-maxed with 31 IVs, colloquially referred to as '6 IV' Pokemon. However, IVs are random and difficult to track; and until Gen 7, it was totally impossible to 'fix' imperfect IVs on your Pokemon. As a result, players use RNG manipulation to get Pokemon with good IVs, especially before Gen 6 (when breeding became a reasonable alternative).
The developers have gradually made IVs easier to access and control. In particular, items like the Destiny Knot and Power items can help you breed high IVs onto Egg Pokemon, and features like the Poke Radar, SOS chaining and Max Raid Battles help you encounter high-IV Pokemon. Since Gen 6, event Pokemon and static encounters have had three guaranteed perfect IVs, and since Gen 7, it has been possible to view a graph of your Pokemon's IVs using the PC. Gen 7 also introduced Super Training, which lets you boost your Pokemon's stats up to perfection using Bottle Caps (though it does not increase the actual IV).
Like EVs, IVs took a different form in Gen 1 and 2. In these old games, IVs ranged from 0 to 15, and the HP IV was generated using the Pokemon's other IVs. Furthermore, IVs were used to determine traits like gender and Shininess; this functionality has been progressively transferred to a different system called personality values (which most people can happily ignore).