Here is a general guideline-- If you live in the Southern US they are all Arbors or Trellises or Patio Covers. Click Pergolas link in the title to see all kinds.
In most of the rest of the world a pergola is a room without walls.
In Australia a Pergola is a roof in the yard and normally has sheet metal applied to reflect the sun.
There are many different variations on the theme.
Generally an arbor is a gate arch or 4 post entrance frame when entering a yard.
This is a gated pergola entrance with trellis screens, (or trelliage or trelliswork) in my humble opinion.