Use a tape measure to get the dimensions of all sides of your patio. Then decide what kind of roof you want. You can make a peaked roof, a slanted roof or a flat roof with just enough slope to let water drain. It's best to build the frame of wood, but you can roof it with roof decking and shingles, galvanized corrugated steel, aluminum panels or plexiglass panels, which will let in some light but block rain. Once you've picked a roofing style and material, use your patio measurements to order supplies. Use 4-by-4- or 6-by-6-inch posts and beams depending on the weight of your roofing material.
Place a 4-by-4 post in each corner, at the edge of your concrete patio, and additional ones on the sides, depending on the length to be covered. Dig holes roughly a third the length of the post, then fill around the posts with concrete. Use a level to be certain the posts are straight and plumb and a tape measure to make sure they are the same height. Nail or screw 4-by-4 beams across the tops of the posts on the longest side; nail or screw 4-by-4 beams to the inside edges of the other sides. Cut 45-degree-angle diagonal braces and mount them between posts and beams at all junctions. Nail or screw these in place.
For a flat or slanted roof, put roof joists between beams on two sides, using metal joist hangers that nail or screw in place. Space them to fit the width of your covering panels or sheathing. Use 2-by-4 or 2-by-6 joists, depending on the weight of your planned roof. For a peaked roof, place roof trusses between beams on the longest sides; you can buy pre-formed trusses at lumber yards and building supply stores. These install with metal truss hangers.
For a shingled roof, nail down oriented strand board (OSB) decking, then cover it with roofing paper and nail on shingles, starting at the bottom and working to the top. For galvanized corrugated steel, aluminum panels or plexiglass panels, nail or screw panels in place starting at the bottom and overlapping joints at least 2 inches. Overlap side seams; some steel and aluminum roofing panels come with interlocking edges, so one panel slides into the next. For a peaked roof, finish with peak caps.