wrong sides together, sew both pieces; add binding, ribbon, or lace to edges.
turn cap to inside (for shower cap - plastic side; for sleep cap - whichever you choose as your inside). Measure and mark 1" from edge (for adults); or 2" from edge for (children); all the way around. Sew along this line, repeat stitch about 1/2" away to form elastic casing.
measure elastic to fit. Cut tiny slit on inside to thread the elastic; secure the ends.