Question: I have a linear regression model for a dependent variable $Y$ based on two independent variables, $X1$ and $X2$, so I have a general form of a regression equation 

$Y = A + B\_1 \cdot X\_1 + B\_2 \cdot X\_2 + \epsilon$, 

where $A$ is the intercept, $\epsilon$ is the error term, and $B\_1$ and $B\_2$ are the respective coefficients of $X\_1$ and $X\_2$. I perform a multiple regression with software (statsmodel in Python) and I get coefficients for the model: $A = a, B\_1 = b\_1, B\_2 = b\_2$. The model also gives me $p$ values for each coefficient: $p\_a$, $p\_1$, and $p\_2$. My question is: What is the null hypothesis for those individual $p$ values? For example, to obtain $p\_1$ I know that the null hypothesis entails a 0 coefficient for $B\_1$, but what about the other variables? In other words, If the null hypothesis is $Y = A + 0 \cdot X\_1 + B\_2 \cdot X\_2$, what are the values of $A$ and $B\_2$ for the null hypothesis from which the $p$-value for $B\_1$ is derived?

Answer: 
The null hypothesis is 
$$
H\_0: B1 = 0 \: \text{and} \: B2 \in \mathbb{R} \: \text{and} \: A \in \mathbb{R}, 
$$
which basically means that the null hypothesis does not restrict B2 and A.
The alternative hypothesis is 
$$
H\_1: B1 \neq 0 \: \text{and} \: B2 \in \mathbb{R} \: \text{and} \: A \in \mathbb{R}.
$$
In a way, the null hypothesis in the multiple regression model is a composite hypothesis. It is "fortunate" that we can construct a pivotal test statistic that does not depend on the true value of B2 and A, so that we do not suffer a penalty from testing a composite null hypothesis.

In other words, there are a lot of different distributions of $(Y, X1, X2)$ that are compatible with the null hypothesis $H\_0$. However, all of these distributions lead to the same behavior of the the test statistic that is used to test $H\_0$. 

In my answer, I have not addressed the distribution of $\epsilon$ and implicitly assumed that it is an independent centered normal random variable. If we only assume something like 
$$
E[\epsilon \mid X1, X2] = 0
$$
then a similar conclusion holds asymptotically (under regularity assumptions).
The $p$-values are the result of a series of $t$-tests. The null hypothesis is that $B\_j=0$, while the alternative hypothesis (again, for each coefficient) is, $B\_j\ne0$

(see here for more details: <http://reliawiki.org/index.php/Multiple_Linear_Regression_Analysis#Test_on_Individual_Regression_Coefficients_.28t__Test.29>)