The four commonest knots used by kite fliers are the larks head, figure of eight, overhand knot, and prussic. These are all shown in the Kite Fliers Knot Gallery.
Also useful for when you only have one end of the line free, are the fisherman's bend and the bowline knot. The fishermans bend is useful for tying a kite line to a fixed object because it's easily undone. Either can be used for tying a bridle line to a towing point on a kite when the other end is already tied, but the bowline knot is possibly more secure, though not necessarily so when using thin or high quality low friction lines.
Knots should never be tied directly in high quality lines such as
Spectra and Dyneema since the breaking strength will be very
considerably reduced. The line should be covered with special widely
available sleaving over the length to be knotted.
The above knots and many others are described or referenced in the Ropers Knot Page.