I'm trying to refer to the coefficients other than the intercept. Is there a word/jargon that refers to coefficients other than the intercept? (I'm currently calling them 'other coefficients', which is mildly descriptive in the context, but not ideal.)
Consider the multiple linear regression model
Here $y$ is the response vector, $X$ is the design matrix with (say) $p+1$ columns (the first column in this matrix is a vector of all ones corresponding to the intercept), $\beta=(\beta_0,\beta_1,\ldots,\beta_p)^T$ is the vector of regression coefficients and $\varepsilon$ is the random error.
Without resorting to vectors, we can write the model as
$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\cdots+\beta_p x_p+\varepsilon$$
In this model with $p$ regressors or predictor variables, the parameters $\beta_j,\,j=0,1,\ldots,p$ are simply called the regression coefficients. In fact, $\beta_j$ represents the expected change in the response $y$ per unit change in $x_j$ when all of the remaining regressor variables $x_i\,(i\ne j)$ are held constant. For this reason, the parameters $\beta_j,\,j=1,2,\ldots,p$ are often called partial regression coefficients. The parameter $\beta_0$ is of course separately called the intercept.
In simple linear regression we have $p=1$ and the regression coefficient $\beta_1$ is simply called the slope.