In most cases, if the effect of x1 on y vanishes when x2 is entered into the regression, there is some kind of non-zero "correlation" between x1 and x2. (If the correlation between x1 and x2 is zero, nothing will happen to x1.). Then the question becomes: Does x1 "cause" x2, or is it the other way around? If x2 comes at a later point in time (i.e. is "caused by x1), x2 is an intervening (or mediating variable) in the process x1-->x2-->y. If x2 precedes x1 in time, in contrast, it is a confounding variable. In this latter case, the relationship between x1 and y would be spurious.

I have a situation quite opposite to this one. I have x1 and x2 that correlate significantly with each other but none of them correlates with y.

However, x1 significantly predicts y in a multiple regression model regardless of whether or not x2 is controlled.

How do I interpret this? @Christer Thrane

If x1 do not have a significant effect on y in a bivariate regression, but have a significant effect once x2 is included in the (now multiple) regression, this is called a suppressor effect. There is some kind of correlation between x1 and x2 that amplifies the effect of x1 on y. (Or vice versa.) In terms of final interpretation, the multiple regression is interpreted in the usual manner :-)