When a car turns, the wheel on the outside of the turn needs to move a greater distance than the one on the inside, as when turning left, the right side wheels will travel the greater distance compaired to the left side. This is acomplished by the diffrential.
A car can still turn without a diff, but it is difficult.
As far as the loose/tight thing, that can be used as a tuning tool. It is best to keep the diff as tight as needed without locking it up compleatly. In the days before slipper clutches, the diff could be adjusted to accomidate diffrent track surfaces. Loosened to help tracktion on a slipery track.