Single-Line Kites by Shape
Here are some key examples of single line kites by shape.

Flat Kites
The lifting surface is a single plane. When configured for flight, the kite is flat when laid on the ground.
Flat kites are usually a simple geometric shape like a diamond. Hexagons, cobras, dragons, and della Portas are all examples of flat kites; many but not all kinds of fighter kites are flat kites (see Fighter Kites).
Flat kites usually require tails for stability, but fighter kites do not require tails.

Bowed Kites
The lifting surface is a single plane, bowed by use of a fixed or variable dihedral or bowed spar.
When configured for flight, the kite is bowed [not flat] when laid on the ground. Eddys, Edos, Bermudas, rollers are all examples of bowed kites.
Rokkaku kites are also bowed kites; see Fighter Kites for more info.

Cellular and Dimensional Kites
Cellular kites are a three-dimensional style of kites that include a wide variety of box kites, snowflakes and stars. The many variants include the Hargrave, the triangular box kite with wings and the Cody, a full box with upturned wings.
Some cellular kites are elaborate in their cell structures and sometimes have wings, fins and/or vanes. A number of cellular kites flown together on a line have the strength to lift heavy payloads.

delta Kites
Delta and delta derivatives kites often have triangular (delta) shapes. They have spars on the leading edges of their wings; these spars can pivot on the spreader bar.
One or more longitudinal spreader spars help these kites keep their shapes. Keels usually are often used but are not always necessary.
In flight, delta kites are typically three-dimensional but they appear to be flat kites when on the ground.

Soft & Flexible Kites
These three-dimensional kites depend on the force of the wind on the kite, not on a rigid frame, to give the kite its shape.
These kites may change their shape depending on the force of the wind. Soft kites like parafoils have no spars.
Flexible or semi-rigid kites like sleds and circoflexes may incorporate spars, with the spars going in one direction or one plane only.