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I have been asked what violates the assumptions of regression analysis, but I don't know how to answer! They said it was either "The error term does not have the normal distribution", "The error term is uncorrelated with an explanatory variable", "The error term has a constant variance", or "The error term has a zero mean". I don't know... can you also explain the answer so I can understand it better?

Thanks!

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Your question is a little broad so I will try to write briefly some of the assumptions statisticians make about the variables used in the analysis. If you need explanation of a particular assumptions, look up CV, and if useful thread not found, post a new question. Consider the regression equation $\hat y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_px_p + \epsilon_i$ and below are the common OLS regression assumptions:

  • Linearity: relationship between dependent and independent variables is linear in nature. You should see from scatter plot of DV vs IV whether the relationship is linear. Various transformations help achieve linearity in case of non-linear relationship.
  • Normality: the variables as well as the unexplained error term, $\epsilon$, are normally distributed (bell shaped). It should be clear from histograms of variables and of error terms (residuals) whether normality assumption holds. A normal probability plot or normal quantile plot of the residuals can be used to check if the distribution of $\epsilon$ is normal. More about normality.
  • Statistical independence of the errors: the error terms, $\epsilon, $ are not correlated with independent variables. Also, there is no correlation between consecutive errors themselves in the case of time series data.
  • Error term has zero-mean: the mean of errors is $0$, i.e., $E[\epsilon] =0$
  • Homoscedasticity (constant variance) of the errors: the error term does not vary over time. A plot of residuals versus predicted values should indicate presence of constant variance. See more about homoscedasticity
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    $\begingroup$ (+1) But statisticians sometimes make some of these assumptions but not others: it can be useful to think about which conclusions of those you might want to draw depend on which assumptions. Normality of the errors, for example, isn't needed for OLS estimates to be BLUE (best linear unbiased estimator). By the way, this is an FAQ, & I'd encourage you, when you recognize an FAQ, to consider linking to other posts that deal with it. [Oh, I see you just did that in your edit while I was writing this comment! :)] $\endgroup$
    – Scortchi ♦
    Commented Jun 3, 2016 at 8:53
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This question is not precise but I think the answer you are looking for is the zero conditional mean assumption that says that E(error | X) = 0. Note: you need strict exogeneity to show that OLS is unbiased (a small sample property) but for consistency (a large sample property) the weaker assumptions of zero mean of the error term and errors being uncorrelated with each regressor are sufficient. Obviously this is not the only assumption: for instance, you also need the full rank assumption for consistency but since you question is very generic, I am unable to provide a complete answer. You might find it useful to refer to an introductory econometrics textbook like Wooldridge, J. (2015). Introductory econometrics: A modern approach. Nelson Education.

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