It means that you play it legato.
Think of how you walk, you put one foot down and after it is in place, you lift up the other foot to move it forward. When you are playing legato, this is what your fingers are doing. The next note is being pressed down before you release the previous note.
Sometimes you play notes non-legato. This is where there is a very small break between one note and the next. You will often do this at the end of one phrase.
You can also play notes staccato or staccatissimo. Staccato (dots) would be played for about 1/2 the length of the note and the let up. Staccatissimo (small wedges or triangles pointing to the note) is about 1/4.
Another articulation form you will see is portato. This is where the notes are dotted as in staccato under a slur. Play the notes for about 3/4 (or just over) of the time.
You play all these this way before adding in the pedal. The damper pedal is more for effect instead of just holding the notes or letting you play legato when you cannot, because when you press the damper pedal, all the strings are allowed to vibrate in resonance with the strings you are playing. Pedal is usually indicated with a Ped (in script) to indicate to press the pedal, and a * to indicate to release it or using a line below the staves to indicate when you press and release. You also have to adjust your pedaling to the sound you are making, so it may not be exactly as written.