The other side of Bob Dylan referred to in the title is presumably his romantic, absurdist, and whimsical one -- anything that wasn't featured on the staunchly folky, protest-heavy Times They Are a-Changin', really.
Because of this, Another Side of Bob Dylan is a more varied record and it's more successful, too, since it captures Dylan expanding his music, turning in imaginative, poetic performances on love songs and protest tunes alike.
This has an equal number of classics to its predecessor, actually, with "All I Really Want to Do," "Chimes of Freedom," "My Back Pages," "I Don't' Believe You," and "It Ain't Me Babe" standing among his standards, but the key to the record's success is the album tracks, which are graceful, poetic, and layered.
Both the lyrics and music have gotten deeper and Dylan's trying more things -- this, in its construction and attitude, is hardly strictly folk, as it encompasses far more than that.
The result is one of his very best records, a lovely intimate affair.