The most decorative roof design is the pergola. This does not provide much shelter. It is an open structure that is more like an uncovered frame used to hold up the roof. Pergolas are beautiful, but do not protect you from rain and direct sunshine, although this structure will still define a space quite well. Most pergolas, unlike other roof structures, extend past the area that it is covering. When building this type of patio roof, consider the distance between the beam and the depth of the beams. Spacing the beams 24" wide will provide half as much shelter as a 12" spacing. The deeper the beam, the more shelter it will provide. To dress up your new pergola, you can use vines or a router on the corners of the beams.
Another option is a vinyl or metal roof. These are the most lightweight type of roof that can be added to your home. A vinyl roof will always protect you from the sun, but not the rain. A metal roof will protect from you from the sun and the rain. You can add color to the roof or stay neutral. These roofs, like all roofs except the pergola, will extend to the edge of the patio that you want to cover. It will define the space as well.
The last couple of options are much more permanent. The first is to add a "lean to" style of roof. This is an angled roof that is constructed to match your existing home. It is usually covered with cedar on the underside and shingles on the top. To really make it unique, consider leaving the framing exposed to gain the look of the pergola from the inside, while maintaining the full protection from the elements. Design is more important with the exposed framing. You will need to plan to use a "2 by" material (2 x 6 is typical) for decking or else the roofing nails will show.
The last type is the hip style roof. It is the same as the "lean to," except the top of the roof will be vaulted. This gives height and size to the structure. It is the most costly roof option, but will look the least like an addition. A hip style roof may not be a viable option depending upon your existing roof. You will need to contact a contractor for this option; he or she can give you more details.