Interpretation of positive and negative beta weights in regression equation I received this elementary question by email:

In a regression equation am I correct
  in thinking that if the beta value is
  positive the dependent variable has
  increased in response to greater use
  of the independent variable, and if
  negative the dependent variable has
  decreased in response to an increase
  in the independent variable - similar
  to the way you read correlations?

 A: In explaining the meaning of regression coefficient I found that the following explanation very useful. Suppose we have the regression
$$Y=a+bX$$
Say  $X$ changes by $\Delta X$  and $Y$ changes by $\Delta Y$.  Since we have the linear relationship  we have
$$Y+\Delta Y= a+ b(X+\Delta X)$$
Since $Y=a+bX$ we get that
$$\Delta Y = b \Delta X.$$
Having this is easy to see that if $b$ positive, then positive change in $X$ will result in positive change in $Y$. If $b$ is negative then positive change in $X$ will result in negative change in $Y$.
Note: I treated this question as a pedagogical one, i.e. provide simple explanation.
Note 2: As pointed out by @whuber this explanation has an important assumption that 
the relationship holds for all possible values of $X$ and $Y$. In reality this is a very restricting assumption, on the other hand the the explanation is valid for small values of $\Delta X$, since Taylor theorem says that relationships which can be expressed as differentiable functions (and this is a reasonable assumption to make) are linear locally.
A: As @gung notes, there are varying conventions regarding the meaning of ($\beta$, i.e., "beta"). In the broader statistical literature, beta is often used to represent unstandardised coefficients. However, in psychology (and perhaps other areas), there is often a distinction between b for unstandardised and beta for standardised coefficients. This answer assumes that the context indicates that beta is representing standardised coefficients: 


*

*Beta weights: As @whuber mentioned, "beta weights" are by convention standardised regression coefficients (see wikipedia on standardised coefficient).
In this context, $b$ is often used for unstandardised coefficients and $\beta$ is often used for standardised coefficients.

*Basic interpretation: A beta weight for a given predictor variable is the predicted difference in the outcome variable in standard units for a one standard deviation increase on the given predictor variable holding all other predictors constant.

*General resource on multiple regression: The question is elementary and implies that you should read some general material on multiple regression (here is an elementary description by Andy Field).

*Causality: Be careful of language like "the dependent variable has increased in response to greater use of the independent variable". Such language has causal connotations. Beta weights by themselves are not enough to justify a causal interpretation. You would require additional evidence to justify a causal interpretation.
