A skill is a bit of practical knowledge possessed by a character. It represents the talent to do something that improves with training, but not something that is intrinsic to the character's class. For example, a sorcerer is a spell caster. Thus, the ability to cast spells is intrinsic to the sorcerer class, and this ability automatically improves with every sorcerer level. On the other hand, the ability to identify magical items is not a defining attribute of a sorcerer; instead, it is a skill (lore). An interesting twist on this is the rogue, since that class is characterized by having more skills than the other classes.
Skills are the characteristics which differentiate members of the same class. This allows a character to be customized to a player's style. For example, a sorcerer who prefers to cast spells in the middle of combat would want to be skilled in concentration, hence be able to cast spells while being pummelled. A sorcerer who prefers to stand back and disrupt enemy spells would want to be skilled in spellcraft, hence be able to identify a spell as it is cast (e.g. before a fireball spell explodes). One might say that skills flesh out a character, resulting in something more than a stale stereotype.
While skills are not considered intrinsic to a class, not all skills are equally available to all classes. Each class has a list of class skills, which are those skills that fall within the expertise of the given class. The remaining skills are either cross-class or prohibited. Cross-class skills fall outside the expertise and typical experience of the given class, but can be improved with extra effort. Prohibited skills fall outside the abilities of the given class and may not be trained in.
Not all skills from Dungeons & Dragons were implemented in Neverwinter Nights, and some skills were adapted to better suit a computer game. (Technically, Neverwinter Nights and the third edition of Dungeons & Dragons were developed in parallel, so some "adaptations" might be more accurately termed "independent evolution".) In particular, the climbing, swimming, and horse riding skills were often requested by players prior to the release of Neverwinter Nights, but, due to technical difficulties and limited development time, those three skills were not included in the original release of Neverwinter Nights. Swimming and climbing were never implemented by BioWare, but the ride skill was included as part of the final patch (1.69).
Skills do not automatically improve when a character gains levels (though some feats, such as bardic knowledge, provide a level-derived bonus.) Instead, a character is given a number of skill points when a new level is gained. Those skill points can be used to buy skill ranks, which represent the character's training in those skills. The number of skill points gained depend on the character's class, intelligence modifier, and race (humans get an extra skill point). At level one (character creation), a character receives four times the normal skill points.
For class skills, ranks cost one point and the maximum number of ranks in a single skill is 3 + character level. For cross-class skills, ranks cost two points and the maximum number of ranks in a single skill is (3 + character level)/2. The determination of a skill being class or cross-class is based on the class being leveled in, and not any other classes the character may have. As a result, the maximum number of ranks in a given skill may go down when a character takes a new class. This is not a problem. What happens in this case is the character will not be able to buy additional ranks in that skill, and previously bought ranks will not be lost.
A player is not obliged to spend all skill points each level-up. Some (or all) skill points can be saved, presumably to be spent at a future level up. The probable motivation for this is to compensate for the lack of half-ranks (so players can save an odd skill point when cross-classing a skill). The more common use, though, is to save skill points until leveling in a different class, in order to spend them on class skills rather than cross-class skills.
Each skill has an associated ability. The ability modifier is added to the number of ranks to produce the character's level of competence in the skill. This number can be further modified by feats, items, and various effects (from spells and the like). Bonuses from items and effects typically stack, to a cap of +50. The final modified number, the skill level, is recorded on the character sheet. (The only time a character's base ranks are displayed is during level-up.)
The descriptions of the skills explain when and how those skills are used. The use of most skills results in a skill check, which is basically a die roll. If the roll plus the character's skill level is high enough, the character successfully used the skill.