Solo means playing alone, without other (human) players to help you.
In pre-Searing, it is possible to complete all quests (except for two that specifically require an ally) solo, but as the story progresses, this becomes increasingly difficult. In post-Searing, the strength and number of enemies in a given area is usually balanced against the maximum allowable party size, so going out solo is guaranteed to be quite a challenge.
Soloing however is all about knowing the types of foes in an area and going out with a set of attributes and skills specifically tailored to counter those foes. Theoretically, any profession can solo, but certain professions and profession combinations may be more suited for soloing certain areas than others. Warriors make a good sturdy base for soloing builds because of their naturally high armor.
For example, a solo-monk relies on enchantments to protect himself and reflect damage back to his foes. As such, he would be most suited to face an area filled with warriors (who do not have healing abilities) and would be most vulnerable in areas with mesmers or necromancers who can remove his enchantments. A Minion Master can quickly flood enemies with his or her undead legion, but cannot use this tactic in areas where the foes do not leave corpses.
In PvP, particularly Guild versus Guild matches, the term solo is used to refer to players in a Split formation who break from the group in singles and sneak in to the enemy base, usually to take out the enemy NPCs. Such a player character is sometimes called a ninja, and the word ninja is also borrowed as a verb to describe this tactic.