Although the terms often are used interchangeably, trellises, arbors and pergolas are distinctly different structures. A trellis is typically a latticework built to support climbing plants or vines. It can be a simple panel attached to the side of a building, or it can be freestanding in a garden or yard.
An arbor usually incorporates a trellis into its structure, creating a tunnel-like passageway of climbing plants. Arbor's have a continuous run of latticework from one side of the "tunnel" to the other, often in an arched shape.
Pergolas, too, are designed to support climbing plants. Unlike arbors, though, pergolas simply have posts supporting a rooflike structure. They're most commonly used to shade a walkway or deck.