Mechanics of Skill Training
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SP are earned over time, whether your character is online or offline, by passively training a skill. This guide exists to give you a better understanding of how exactly the game goes about this.
The amount of SP you earn over time (and hence how long a skill will take to train to the next level) is dependent on your attributes. Specifically, it is dependant on the primary and secondary attributes that are used in training the skill. For example, gunnery skills use Perception (primary) and Willpower (secondary) as attributes in determining how long a gunnery skill will take to get to the next level. Having higher Per and Will will make all Per/Will skills train faster. Industry skills, on the other hand, use Intelligence and Memory, so higher Int/Mem will result in faster Industry training. The primary skill attribute is twice as important as the secondary skill attribute for calculating training time. The formula is
Skill points per hour = 60 * Primary Attribute + 30 * Secondary Attribute or, from the perspective of training time
The number of minutes to train the skill to a given level = SP Needed  Current SP of skill / (Primary Attribute + Secondary Attribute/2)
Thus, additional levels in an attribute will bring diminishing returns, increasing all attributes from 10 to 11 will decrease skill training times by roughly 9.1% (SP/hr +10%), but increasing both attributes from 11 to 12 will only decrease times by 8.3% (SP/hr +9.1%)
As you will see in EVEMon, your ability in a particular attribute = (Your base ability in that attribute + your levels in the relevant learning and advanced learning skills (eg. Instant Recall and Eidetic Memory) + the bonus from any implants you have) * (1 + 0.02 * your level in the "Learning" skill)
As to question of how many skill points a level in a skill "costs", each level of a skill takes ~5.65 (square root of 32) times as much SP than the preceding level. The rank of the skill is a training time multiplier. The formula is:
Where r is the rank of the skill, and n is the level; the brackets mean "round up". For example, training a rank 2 skill to level 4 takes ceil(250*2*sqrt(32)^{3}), or 90,510 SP. A simpler way to quickly calculate the SP at a given level:
 Level I = 250*rank SP
 Level II = 1414*rank SP
 Level III = 8000*rank SP
 Level IV = 45255*rank SP
 Level V = 256000*rank SP
Remember, because the formula is SP Needed  Current SP, the time it takes to go from IV to V is a bit less than the time it takes to go all the way to V; ex. a Spaceship Command (rank 1 perception/willpower skill) takes 118.5 hours (specialized: 28 perception and 16 willpower) to train from 0 to V, but only 97.5 hours to train from IV to V  compared to the time from 0 to IV, which is a mere 21 hours. Meanwhile, a Racial Battleship skill would take 8 times that (780 hours from IV to V, 948 from 0 to V: 32.5 and 39.5 days respectively), yet from 0 to IV is only 167.6 hours (one week).
For newbees, if you don't have to have the skill at V due to fitting, module, or hull requirements (or Thermodynamics), it's better to train it only up to IV until you have time to spare and want to push for that last 20% of the bonus.
As another example (speaking approximately) a rank 1 skill might take 15 minutes to get to the first level. It would then take 5.65 x 15 for level 2, or 83.4 minutes (about 1 hour and 20 minutes.). Level 3 would take 5.65 x 83.4 minutes (463, or about 7 1/2 hours) and so on. If the rank of the skill was 2, you would double all of these times. Rank 3 would triple them, and so on.
 Note that skills no longer train when your account has lapsed.