Descriptions of how each articulation is done cannot be given here, because they depend too much on the particular instrument that is making the music. In other words, the technique that a violin player uses to slur notes will be completely different from the technique used by a trumpet player, and a pianist and a vocalist will do different things to make a melody sound legato. In fact, the violinist will have some articulations available (such as pizzicato, or "plucked") that a trumpet player will never see.
So if you are wondering how to play slurs on your guitar or staccato on your clarinet, ask your music teacher or director. What you will find here is a short list of the most common articulations: their names, what they look like when notated, and a vague description of how they sound. The descriptions have to be vague, because articulation, besides depending on the instrument, also depends on the style of the music. Exactly how much space there should be between staccato eighth notes, for example, depends on tempo as well as on whether you're playing Rossini or Sousa. To give you some idea of the difference that articulation makes, though, here are audio examples of a violin playing a legato and a staccato passage. (For more audio examples of violin articulations, please see Common Violin Terminology.)