Your skills in Eve are the most important part of developing your character. They are what gives you the ability to use new modules and ships, and to increase your effectiveness at a wide range of actions. They can help a miner mine more ore faster, help a fighter shoot faster and more powerfully, and help a manufacturer lower his wastage of materials to near nothing.
First, an introduction:
You will have started Eve by creating a character. Here you'll have chosen a few things about that character--what school they went to, what they specialized in, etcetera--that will determine their starting skills. Maybe you'll start with Navigation IV or Frigate III, or maybe you'll start with Mechanic at a nice level. But the important thing is to make sure that you either have a good balance of skills, or a good skillset to help you in a particular field--depending on what you want to do in Eve. You can buy skillbooks from the Market*, then go to your Items, right-click the skill and pick Train Skill. Then the skill will be on your Character Sheet list of skills, under one of the drop-down categories. You can only train one skill per -account- at any given time. Switching skill-training is okay; the game will hold your place on partially-trained skills.
*This means that you could train up several different fields if you like--which means that, during character creation, getting good attributes is more important than anything else. Generally, Caldari Archura characters are known for having the best attributes for PvP play, regardless of which race's ships you train for. Play around with the character creation, if you don't already have a main character, and see what you can get. See the Attributes section below for more.
Rank and Level: A skill can be trained to level V (5). The effectiveness of a skill may increase each level, or perhaps it allows certain modules to be used at higher levels. For example, Mechanic creates a 5% bonus to your ship's base structure (hull) HP every level, so at level V you'll have 25% more HP. However, other skills or modules not listed on the Mechanic information tab (which you can reach by right-clicking and Show Info) may require a certain level of Mechanic to work. For example, someone who wants to train the skill Assault Ships will have to train Mechanic to level V first. As another example: Afterburner level I enables use of standard frigate Afterburners. This is listed not under the Skill info but under Afterburner's Required Skills. It sounds complicated but it's really very simple. All you need to worry about is looking at an item you want, checking the Required Skills and then going to your Character Sheet's Skills tab and training the corresponding skill (or buying it first, if it isn't listed--skills can be purchased at any University dock, along with a lot of other docks).
Now, if you look at any given skill, you'll see a number in parenthesis. Taking Mechanic again, you'll see Mechanic Rank (1). The rank is simply a time multiplier. A Rank 1 skill is a basic skill. A Rank (2) skill will take 2 times as long to complete. The larger the rank, the longer a skill takes to train, it's as simple as that. The actual amount of time training will take depends on your Attributes, as follows.
Learning Skills and your Attributes:
Learning skills are very important. If you were to go to your Character Sheet and click on the Attributes tag, you'd see numbers indicating your levels of Intelligence, Perception, Charisma, Willpower and Memory. Different skills' training times depend on different attributes. The higher your Memory and Intelligence, for example, the faster your Learning skill goes up. Gunnery, on the other hand, depends on your Perception and Willpower. You can check what skills depend on which attributes by right-clicking the skill and choosing Show Info, then looking at the Attributes tab. You can raise your attributes one of two ways:
1. Put in Implants. These are usually better for more advanced players; implants raise a particular attribute from 1 point to 5 (depending on the particular implant) but are extremely expensive, cannot be removed for safekeeping and are lost if the wearer's pod is destroyed. Implants can be a good temporary learning boost, but relying on them wholly, and ignoring learning skills, is not at all recommended.
2. Learning Skills. This is highly encouraged by most older players to be the first thing to train. The tier 1 learning skills are Learning (which only increases learning speed, no attributes), Analytical Mind (raises Intelligence by 1 point per level), Empathy(does the same for Charisma), Instant Recall (memory), Iron Will (Willpower) and Spatial Awareness (Perception). If you train these up to level 4 or 5, you'll have a huge decrease in the amount of time any given skill takes to train. You could then (as of the basic skill's level 4) move on to the tier 2 Learning Skills, which are rank (3) rather than (1)--but you'll learn more about those as you progress in Eve. Training Learning Skills can be very boring, but speed things along quite a bit in the long run; it's best to alternate between Learning and other skills to keep yourself happy!
SkillPoints
SkillPoints, or SP, are visible as your total SP atop your Skillsheet or as an individual skill's accumulated SP, visible under the skill itself. Someone with Analytical Mind trained to level V, for example, will show SP: 256,000 beneath it, meaning they have 256,000 skillpoints there. Skillpoints are often used as a general gage to how advanced a player is in Eve; the more SP, the more highly-trained their skills tend to be. Some of the more advanced players will often have fifty million skillpoints or more--but don't be discouraged. You can be a good player in Eve without amassing huge numbers of skillpoints! That said, note that some corporations require a minimum number of SP of new recruits. This is usually to ensure not that the person is a great player, but that they are dedicated to the game and won't leave a week later.
Clones
A note: this will be covered further in the general Eve guide. However it deserves a mention here: you need to keep an updated clone close to where you are operating at any given time. "Updated" means that the clone must have the ability to "hold" more skillpoints than you currently have. If you get podkilled, and you have (for example) 100,000 skillpoints, but your clone was only good until 90,000, you'll lose 10% of that difference (the difference being 10,000). Therefore you'll have to retrain some of your skills to get back the lost 1000 skillpoints! This isn't a lot, but most players are spawned with just under a million sp, and hit a million in a short time. So be certain that you always have an up-to-date clone!
Skills for Careers
So! You want to be a Miner? You'll want to train up Mining, and Industry. You'll also want to train up your Drones skills so that you can use drones for protection (or mining drones, although they're sometimes considered rather inefficient). You'll eventually need to train your skills up for different ship types so that you can use mining cruisers and eventually mining barges!
Fighters, you'll need Gunnery skills, along with Missile skills (Missiles are actually more important if you're a Caldari character), plus Navigation skills. Weapons Upgrades are important, and Mechanic, Engineering and Electronics will all play important roles in allowing you to fit your ship with good weaponry and modules. Your Spaceship Command is important: you'll want to be able to fly a Cruiser fairly soon after beginning.
Production folks will want Manufacturing and Refining skills and their Industry skills skills up.
A few basic skills important for ship flying:
To boost your ship's CPU output, train Electronics up.
To boost your ship's Power output, train Engineering up.
Train Afterburners to be able to use afterburners--these are invaluable, regardless of your Eve career path.
Train Repair Systems so you can use Armor Repairers.
More details on specific Eve careers are present in their own guide on the Guides page. Good luck, and happy training!