Errors in variables are measurement errors which increase the estimation variance (error in the dependent variable) or bias the regression coefficients towards zero (error in the independent variables). Consider measurement error in the independent variable and suppose you want to regress: $$Y = \alpha + \beta X + u\newcommand{\Var}{{\rm Var}}$$ but the independent variable has some error in it (e.g., due to wrong coding or typos) such that we have $X = \tilde{X} + e$, where $X$ is uncorrelated with both $u$ and $e$. Hence in the previous regression we have \begin{align} Y &= \alpha + \beta (\tilde{X} - e) + u \newline &= \alpha + \beta \tilde{X} + \varepsilon \end{align} where $\varepsilon = u - \beta e$. It can then be shown that the estimated coefficient $\widehat{\beta}$ is biased towards zero for which the downward bias is given by the noise-to-signal ratio: $$\widehat{\beta} = \beta \left( 1-\frac{\Var(e)}{\Var(X)+\Var(e)} \right)$$ This so-called attenuation bias is decreasing in $\Var(X)$ (the strength of the signal) and increasing in $\Var(e)$ (the amount of noise or measurement error). A common tool for circumventing errors in variables is the use of GMM and instrumental variables methods.
For the case of measurement error in the dependent variable the only consequence is that the overall regression error increases, hence weakening the power of statistical testing procedures. Suppose we measure $\tilde{Y} = Y + e$, then the regression $$\tilde{Y} = \alpha + \beta X + (u + e)$$ includes the measurement error in the overall regression error.