I have read in the abstract of this paper that:


  "The maximum likelihood (ML) procedure of Hartley aud Rao is modified by adapting a transformation from Patterson and Thompson which partitions the likelihood render normality into two parts, one being free of the fixed effects. Maximizing this part yields what are called restricted maximum likelihood (REML) estimators."


I also read in the abstract of this paper that REML:


  "takes into account the loss in degrees of freedom resulting from estimating fixed effects."


Sadly I don't have access to the full text of those papers (and probably would not understand if I did).

Also, what are the advantages of REML vs. ML? Under what circumstances may REML be preferred over ML (or vice versa) when fitting a mixed effects model?
Please give an explanation suitable for someone with a high-school (or just beyond) mathematics background!

As per ocram's answer, ML is biased for the estimation of variance components. But observe that the bias gets smaller for larger sample sizes. Hence in answer to your questions "...what are the advantages of REML vs ML ? Under what circumstances may REML be preferred over ML (or vice versa) when fitting a mixed effects model ?", for small sample sizes REML is preferred. However, likelihood ratio tests for REML require exactly the same fixed effects specification in both models. So, to compare models with different fixed effects (a common scenario) with an LR test, ML must be used.

REML takes account of the number of (fixed effects) parameters estimated, losing 1 degree of freedom for each. This is achieved by applying ML to the least squares residuals, which are independent of the fixed effects. 
