This work concerns receiver design for light emitting diode (LED) communications where the LED nonlinearity can severely degrade the performance of communications. We propose extreme learning machine (ELM) based non-iterative receivers and iterative receivers to effectively handle the LED nonlinearity and memory effects. For the iterative receiver design, we also develop a data-aided receiver, where data is used as virtual training sequence in ELM training. It is shown that the ELM based receivers significantly outperform conventional polynomial based receivers; iterative receivers can achieve huge performance gain compared to non-iterative receivers; and the data-aided receiver can reduce training overhead considerably. This work can also be extended to radio frequency communications, e.g., to deal with the nonlinearity of power amplifiers.