I'm a statistics newbie, taking an online course, and the term "re-randomization" was thrown in without explanation. What does it mean?
Rerandomization, as used by Morgan and Rubin (Annals of Statistics 2012), is a form of restricted randomization. Suppose you had all of your subjects available for randomization, e.g. you are randomizing 20 hospitals into two arms. You would like the two arms to be similar in terms of important covariates, e.g. hospital size, teaching hospital, etc. If you had several covariates, there is a nontrivial chance of imbalance in one or more covariates. In rare but serious occasions, a covariate could be highly imbalanced. If you could check your randomization for balance prior to starting your intervention and found imbalance, you could redo your simple randomization until the first time you got a randomization that had satisfactory balance.
Your statistical tests that do not account for the fact that you used rerandomization will be conservative, e.g. your two-sample t-test will have a p-value that is a little too large on average so the Type I error for this test is a little too low and the Type II error is higher than it has to be. However, a test that adjusted for the covariates you considered when deciding whether to accept a randomization, e.g. a multivariate regression that adjusted for those covariates in the model, will have approximately the correct standard error and "better" error rates.
Rerandomization is one form of restricted randomization, but there are several viable options. Other methods include stratified randomization, matched randomization (see Greevy, Lu, Silber, Rosenbaum Biostatistics 2004; Kapelner and Krieger Biometrics 2014; and others), and minimization (see Pocock and Simon Biometrics 1975; and many others). Matched randomization and minimization have the advantage that they can be used with sequential entry trials, where patients arrive one at a time or in small batches, and they scale nicely in terms of the sample size and the number of covariates, as opposed to ordinary stratified randomization which has trouble performing well with more than a few covariates. Any form of restricted randomization typically offers big gains over simple randomization.