In some literature, I have read that a regression with multiple explanatory variables, if in different units, needed to be standardized.  (Standardizing consists in subtracting the mean and dividing by the standard deviation.)  In which other cases do I need to standardize my data?  Are there cases in which I should only center my data (i.e., without dividing by standard deviation)? 
You have come across a common belief.  However, in general, you do not need to center or standardize your data for multiple regression.  Different explanatory variables are almost always on different scales (i.e., measured in different units).  This is not a problem; the betas are estimated such that they convert the units of each explanatory variable into the units of the response variable appropriately.  One thing that people sometimes say is that if you have standardized your variables first, you can then interpret the betas as measures of importance.  For instance, if $\beta_1=.6$, and $\beta_2=.3$, then the first explanatory variable is twice as important as the second.  While this idea is appealing, unfortunately, it is not valid.  There are several issues, but perhaps the easiest to follow is that you have no way to control for possible range restrictions in the variables.  Inferring the 'importance' of different explanatory variables relative to each other is a very tricky philosophical issue.  None of that is to suggest that standardizing is bad or wrong, just that it typically isn't necessary.  

The only case I can think of off the top of my head where centering is helpful is before creating power terms.  Lets say you have a variable, $X$, that ranges from 1 to 2, but you suspect a curvilinear relationship with the response variable, and so you want to create an $X^2$ term.  If you don't center $X$ first, your squared term will be highly correlated with $X$, which could muddy the estimation of the beta.  Centering first addresses this issue.  



(Update added much later:) An analogous case that I forgot to mention is creating interaction terms.  If an interaction / product term is created from two variables that are not centered on 0, some amount of collinearity will be induced (with the exact amount depending on various factors).  Centering first addresses this potential problem.  For a fuller explanation, see this excellent answer from @Affine: Collinearity diagnostics problematic only when the interaction term is included.  
