Skills

Skills in SAO are abilities or features that are given to a player. They function in a wide variety of ways: stat boosts, sword skills sets, support passives, or non-combat capabilities to improve a player's in-game power. In general, skills must be equipped in available slots to be useable. They may be trained up to level 1000, at which they are considered «Completed» or maxed. There is rumored to be an unlimited number of skills in Sword Art Online. 
There are various types of skills in SAO, including: weapon skills, support skills, non-combat, extra skills, unique skills, outside system skills (OSS) and sword skills. 
Each skill type has a different defining aspect that separates it from the others. The following lists will define the category before listing corresponding skills. Listed skills with asterisks next to them can belong to another category. 
== Progression ==
Generally skills are leveled up by continual usage. That means that if a skill is a weapon skill, the correct weapon must be fought with and sword skills must be used. Just like stats, skills are boosted every 50 levels or so, increasing proportionally in power and other factors. That also means that if two users have the same skill, by level 1000 those two skills will be equivalent in every way. However, there are ways to allow skills to differentiate into higher "ranks" or tiers.
Skills are split into ranks ranging from 0-3. Rank 0 skills, or regular skills, are those that can be easily acquired at the beginning of the game with no requirements other than a short conversation with the skill giver. Rank 1 skills are considered "extra" skills and usually have some sort of quest or requirement. Rank 2 skills are also extra skills, but they require higher fulfillments. Rank 3 skills are considered "unique" skills with large requirements, specificity to a single player, and are the rarest of the ranks. As each rank increases, power is increased. That means that a rank 0 skill can be beaten by a rank 2, and a rank 2 skill can be beaten by a rank 3 skill. Rarity also increases with rank as well as necessary requirements.
Some skills can follow a tree or path. This means that they actually progress from rank 0 to rank 1 to rank 2 and so on. Some skills only progress part way; others start at a higher rank and progress from there. Progression is not necessarily singular either. This means that a rank 0 skill might progress to a certain rank 1 skill, or it may progress to a different rank 1 skill. Or, the rank 0 skill might jump up to a rank 2 skill.
A good example of skill rank progression is the rank 2 skill Eagle Sense. This skill originally started off as Detection, advancing to Eagle Pulse, and then finished off with Eagle Sense. It moved upwards linearly and did not jump. An example of rank jumping is Two-Handed Straight Sword to Colossal Pierce to Blade Hawk. 
Not all skills have progressions. Most do, but not all.