NB-regression and linear regression are different things.

linear regression is performed to estimate a linear functional relationship between two continuous (metric) variables (a "response" ["dependent", "Y"] and a "predictor" ["independent", "X"] variable), and the "best fit" is found by maximizing the likelihood of the data based on the normal (gaussian) probability model.

NB-regression is performed to estimate a functional relationship between a count variable as response and a metric predictor. Here, the relationsship between the "linear predictor function" (often symbolized with the Greek eta) and the predictor should be linear. The "best fit" is found by maximizing the likelihood of the data based on the negative binomial probability model.

Hi Jochen Wilhelm, thank you for your response.

Besides linearity assumption, does NB regression need other assumptions such as no autocorrelation, no heteroscedasticity like in linear regression? What is distribution of error?

Hi

The distribution of your data is assumed to be negative binomial, typically with the expected value of the dependent variable given as a linear function of your predictor variables. That leaves one other parameter, a shape parameter which  as Jochen says you will typically estimate by maximum likelihood.

You can think of the shape parameter as being similar to the variance in a normal linear regression problem. Indeed, with shape parameter r and mean m, the variance of the response is m+m^2/r. Notice that the variance automatically varies with the mean (unless r tends to infinity and yoiu are left with a Poisson model). You are not assuming homoskedasticity, but you are (typically) assuming a constant shape parameter r.  

In NB regression you are usually assuming that observations are independent variables. You are just as prone to problems with time dependence as with normal linear regression. NB regression is making different, but equally restrictive assumptions.

There is no reason why you can't build models which generalise away from these assumptions, but at hat point you need statistical help. 

So I am not absolutely shure what you mean with the "assumption of no auto-correlation". To my understanding, auto-correlation can exists only if you either have non-independent data (and independence is the more basic assumption that is always part of any statistical model - except when the dependency-structure is explicitly modelled, like e.g. in mixed-effect models), or if there is a confounding variable with a periodic influence on the response (like circadianian changes etc.). If you face (unwanted) "auto-correlation" then either your model or your sampling is wrong.

Mervin answered well your question regarding heteroscedasticity.

And with "distribution of error" I mean the probability model that is used to quantify our expectations about the deviations between the model prediction and the measures values (the "residuals").

The most important assumption of a Negative Binomial model is the overdispersion of the dependent count variable. That is, the variance of the dependent (y = count variable) variable must be way greater than the mean.