Your character's proficiency in KOTOR's skills will to a large extent determine the path you take through the game. Most soldier characters will use skills only rarely, since most skills are related to non-combat activities, so soldiers will generally have to rely on the talents of NPC teammates to make their way past obstacles. There's never a guarantee that you'll have an NPC along with the right skill when you need it, though, so the choices you make with your PC's skills will still have a great impact on your experience as you play KOTOR. Scouts, and especially scoundrels, will need to hone multiple skills to make up for their lack of raw combat ability.
The number of skills characters get to choose depends on their class, as well as their intelligence modifier. Scoundrels are the most skillful of the KOTOR classes, with scouts coming close behind, while soldiers, rarely the smartest people in the galaxy, lag far behind. You'll be able to select a few skills when you create your character, and you'll get to further enhance your knowledge at each level-up.
Skills, like most everything else in the game, are based upon simulated rolls of a 20-sided die, so even a high level of proficiency in a skill won't always protect you from failure. Most skill-related tasks in the game are assigned a difficulty check (DC) of a number usually between 10 and 30, with higher numbers representing the more challenging scenarios. When you make a skill check, you take your rank in that skill, add or subtract the appropriate modifier for the governing attribute to find your modified skill rank, and then, depending on the circumstances, the game either adds a random number between one and 20 to that rank or simply adds 20 to determine whether or not you succeeded at your task.
The difference between the dice roll and the simple addition of 20 is known as the "take 20" rule: if your character has a theoretically unlimited amount of time to pick a lock, for instance, then their chance of success are going to be much better than that of a character who's attempting to pick the same lock while dodging blaster fire. For the purposes of the game, demolitions and security are the only two skills that are consistently subject to the "take 20" rule, so if you wish to disable a mine, or pick a lock, you'll want to make sure you make your attempt outside of combat.
If the final number that the game generates is equal to or above the DC of the task, then you've accomplished your mission; if it's below, then you failed in some way. Some tasks will allow you to try them multiple times; many, most notably those relying on the persuasion skill, are one-time-only opportunities.
It's important to consider, when making a character, that not all classes are equally adept at all the skills. A soldier is obviously not going to be as useful as a scoundrel when you're confronted with a locked door, and by the same token, you probably wouldn't want a scoundrel looking you over when you need to have a combat wound treated. These congenital deficiencies are represented in-game by the notion of "cross-class skills." Each class has certain skills that they excel at; these are listed in the Class profiles section. These skills are purchased with skill points at a one-to-one ratio. Unfortunately, the cross-class skills, i.e. those that your character just wasn't trained in, are difficult to learn, and are thus much less efficient to improve, requiring two skill purchase points for each single-point upgrade to the actual skill.
Lastly, there is a limit to how high you can raise your rank in a skill. For class skills, this limit is your experience level plus three, resulting in a maximum limit of rank four when you begin the game. For cross-class skills, the limit is half of the limit for your class skills, rounded down, so that, at experience level 4, your upper limit for a cross-class skill would be rank three (experience level four + three equals seven, divided by two is three and a half, which is rounded down).