Select a high-brightness diode or several high brightness diodes. The data sheet will have information about electrical power requirements, the mechanical and thermal interface and the light output distribution.
Compare the total output and light distribution with the requirements for the fixture. Add additional LEDs as necessary to reach the desired illumination level.
Design or obtain drive electronics that will provide the necessary electrical power to the LEDs. Note that traditional dimming will almost certainly not work with the LED, so particular care will need to be taken with the electronics, if dimming is needed.
Mount the LED to a heatsink. The importance of this step cannot be overemphasized. LEDs are electronic chips and, like computer chips, they will overheat unless care is taken to maintain their temperature.
Obtain, or design, reflectors, lenses or diffusers to convert the light output from the LED to the light desired for the application. It's possible to skip this step, or just use a parabolic bowl reflector, but it's very likely the design will end up throwing light away --- putting it where it's not needed. Light fixture designers use (or should use) complex, nonlinear ray-tracing software programs to evaluate and optimize their designs.
Assemble the components in a mechanical package that fits the need for the application. LED lighting manufacturers spend a lot of time integrating the heatsink, electronics and optics into a single, attractive package.