Maximum Likelihood Estimator (MLE), MMSE and LS - Are All of Them Regressor, Estimator and Predictor?
Let's see:


Maximum Likelihood (ML) - A Method to estimate parameters (MLE - Maximum Likelihood Estimator). Given a PDF of samples, what is the parameter value which maximizes the probability of the observations.  
Least Squares (LS) - Least Squares model for estimating parameters. Sometime coincides with the MLE (Usually in the case of Gaussian Noise).
Minimum Means Square Error (MMSE) - Bayesian Estimator based on Loss Function which is the second moment of the error.


All of them are estimators in a model (PDF of Parameters in the ML, Numeric Model in the LS or Bayesian Model for the MMSE).

Predictor is a term for predicting future data based on given data. Any of the method above can do that with different assumption of the model.
Regressor is a term for predicting model parameters from a given data. Again any of the above can be used to do so.

So you need to separate the means you use (Type of estimator and model) for the object you try to achieve (Predict, Regress, Smooth, etc...).