A lean-to shed is attached to a wall or roof of a house, garage or outbuilding with a roof that slopes away from that structure. Ideally, it should be shingled to match the existing roof. Measure the shed roof with a tape measure to determine how many shingles will be needed. They typically are sold in bundles to cover about 100 square feet. The standard style of asphalt or composite shingle is called three-tab because each shingle has one straight edge and one edge with three tabs to overlap a lower shingle.
Install metal drip edge flashing on the three open edges of the roof. Drip edge is bent at 90 degrees with a flat side to go on the roof and a side to go down the edge of the roof with a lip on it to force water away from the roof. Place the drip edge on the bottom of the roof first, starting at one corner. Run a bead of roofing cement under it and nail it with a hammer and flashing nails. Overlap sections if needed in a direction away from any prevailing wind. Cut the drip edge if needed, using tin snips.
Lay asphaltic roofing membrane or tar paper horizontally over the roof sheathing and drip edge. Staple it to the sheathing. Overlap a second row of paper by at least 6 inches. Run drip edge up the sides of the roof over the roofing paper, starting with an overlap at the corner of the bottom strip. Nail drip edge to the roof sheathing about every foot. Run it up flush with the top of the roof.
Put bent metal flashing at the top of the roof, where it joins the lean-to building. Run this down the roof and up the siding or roofing of the adjoining structure, ideally where wood siding or shingles were removed to attach the lean-to shed. Put a bead of roofing cement at the top of this flashing piece. Nail it about every foot. Overlap the side edging from the top.
Snap horizontal and vertical chalk lines on the roof paper to make straight guides for shingles. Cut off the tabs with a utility knife on enough shingles to make one row across the bottom of the roof. Nail those shingles on upside down so the even finished edge is at the bottom of the roof. Nail this row at the top of the shingles, using nails and spacing them as recommended by the shingle manufacturer.
Add a second row of shingles with the tabs overlapping the first. Follow the manufacturer's recommendations on staggering shingle courses so seams don't align; typically cut 6 inches or one tab off an end to start the second row and repeat that with the third row. Return to full shingles on the fourth row, so only every third row of shingles will align (this will vary with the depth of the roof to be covered).
Run shingles to the top of the roof. If existing roof shingles or wood siding were removed to attach the shed roof, run the lean-to shingles up under the old shingles or siding line and reattach siding or shingles to make a watertight fitting. If the shed roof was attached over existing shingles or siding, bend the last row of shingle up to the top of the flashing, nail it in place and seal it with roofing cement.