Both MLE and REML are designed to estimate the parameters of a statistical model. A statistical model simply shows the relationship between variables; the parameter is the unknown number or characteristic the researchers are collecting and analyzing data to determine. For example, if a researcher wants to identify the average IQ of college students in the United States then that is the parameter of the statistical model. Since researchers could not give IQ tests to every college student in the country, they must estimate their parameters from a smaller amount of collected data.
MLE stands for Maximum Likelihood Estimation. The concept arose from the work of R. A. Fisher in the 1920s and says that researchers should look for the probability distribution that will be most likely to fit the collected data. This means that based on data from a portion of the studied population you can estimate what that data infers about the entire population. When using statistical analysis software, multiple parameters are compared by the software to determine which is the most likely to match what the data is showing.
REML stands for Restricted Maximum Likelihood. REML is actually a type of MLE and, therefore, is designed to make predictions about parameters based on collected data. This method arose from the work of M. S. Barlett in the late 1930s. It differs from MLE in that the estimations are not based on all of the data collected but on contrasts identified in the data. Probability distributions are based on these contrasts instead of on the full set of collected data. REML does a better job of making estimations about effects than does MLE.
REML is often used with linear mixed models. These statistical models attempt to show the relationship between more complex data. The models sometimes include random effects, clustered data or longitudinal data. On the other hand, MLE works best with simpler data and normal distributions.