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I've read in Abdi (2003) that

When the independent variables are pairwise orthogonal, the effect of each of them in the regression is assessed by computing the slope of the regression between this independent variable and the dependent variable. In this case, (i.e., orthogonality of the IV’s), the partial regression coefficients are equal to the regression coefficients. In all other cases, the regression coefficient will differ from the partial regression coefficients.

However, the document did not previously explain what the difference between these two types of regression coefficients is.

Abdi, H. (2003). Partial regression coefficients. In Lewis-Beck M., Bryman, A., Futing T. (Eds.) (2003) Encyclopedia of Social Sciences: Research Methods. Thousand Oaks, CA: SAGE Publications.

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"Partial regression coefficients" are the slope coefficients ($\beta_j$s) in a multiple regression model. By "regression coefficients" (i.e., without the "partial") the author means the slope coefficient in a simple (only one variable) regression model. If you have multiple predictor / explanatory variables, and you run both a set of simple regressions, and a multiple regression with all of them, you will find that the coefficient for a particular variable, $X_j$, will always differ between its simple regression model and the multiple regression model, unless $X_j$ is pairwise orthogonal with all other variables in the set. In that case, $\hat\beta_{j\ {\rm simple}} = \hat\beta_{j\ {\rm multiple}}$. For a fuller understanding of this topic, it may help you to read my answer here: Is there a difference between 'controlling for' and 'ignoring' other variables in multiple regression?

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    $\begingroup$ That is a great explanation of the cited text, I would add, however, that I have never in the reporting of results from multiple-predictor-models seen or written about the coefficients as "partial" coefficients. We will sometimes discuss the differences between bivariate and multivariate estimated coefficients (adjusting for confounding variables, for instance, may change the direction or magnitude of the findings). $\endgroup$ – AdamO Jun 14 '16 at 15:10

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