You can try this procedure to reset the iPod shuffle
Then, shut down your computer (power off) and start it up, with iPod shuffle NOT connected. Run iTunes. Then, try connecting the iPod shuffle again.
If that iPod is less than one year old, it is under warranty. You can contact Apple (or make an appointment at an Apple Store Genius Bar) to see about getting it replaced (if it's broken).
Otherwise, two likely possibilties are that (1) the cable used to connect the iPod to computer is bad, or (2) the connector on the iPod has gone bad.
Which iPod shuffle model is it? The document above with reset procedure shows the four models.