Hi. In a multiple regression, the beta coefficients are technically the same as the partial derivatives with respect to that variable, holding all other independent variables constant. So say I have 3 independent variables, X1, X2, X3.
The beta coefficient B1 of X1 is the expected change in Y(our dependent variable) for each one unit change in X, holding X2 and X3 constant. HOWEVER when your variables are in standardized form, the interpretation is a litte different. With standardized variables, a 1 standard deviation change in X1 equals B1 (the beta 1 coefficient) standard deviations in Y. We standardize our coefficients primarily because it allows us to directly compare the beta coefficients. Normally if we don't standardize the coefficients, each variable must be interpreted on the basis of its own units of measurement. Regardless, standardizing variables will not affect whether or not the coefficients are significant.
A negative beta coefficient means that a 1 unit positive standard deviation change in X is expected to result in a negative beta coefficient change in Y. So if your beta is, say, -3, a 1 unit standard deviation change in X is expected to result in a -3 standard deviation change in Y. It's very similar to the slope (it is the slope of our regression line, keeping all other variables constant).
DO NOT ignore the negative sign! It means your two variables (x,Y) are negatively associated/correlated. However, if you notice that the beta coefficient seems to change sign as you add or subtract other X variables, you may have a problem called multicollinearity, where the X variables are highly correlated with each other. This is especially true if the Betas correlate and the R-Square is high (R-Square tells you how much of the variance in Y the model explains). Normally the beta coefficient should not change sign as you add more independent variables. What is the correlation between X and Y? That is, if you simply doa correlate procedure (which is the same as a simple regression), is the sign still negative? If it is, then that's good, it means the two variables are likely negatively associated.
A negative or positive value on the beta coefficient SAYS NOTHING about significance. The only way to calculate significance is to divide the beta coefficient by its standard error. Usually, the rule of thumb is that any value (t-score) greater than 2 is significant at the usual 95% level of significance.
Email me if you need further clarification. Never ignore the sign of the coefficients. It tells you how one variable changes with respect to another.