Staccato is shown by a dot over or under the note on the opposite side to the stem. Staccato notes are often played at half their written value.
Legato is shown by a curved line above or below the notes on the opposite side to the stem. Legato notes are played with no gaps in between the notes. They are played in a smooth and connected manner; this is often called a slur.
Detaché is often shown as a legato slur and staccato dots however sometimes it is shown with no symbol at all. The notes are a little more connected then legato and a little smoother than staccato,