I am confused about the null hypothesis for linear regression.
The issue applies to null hypotheses more broadly than regression
What does that translate to in terms of null hypothesis?
You should get used to stating nulls before you look at p-values.
Am I rejecting the null hypothesis that the coefficient for that variable is 0
Yes, as long as it's the population coefficient, ($\beta_i$) you're talking about (obviously - with continuous response - the estimate of the coefficient isn't 0).
or am I accepting a null hypothesis that the coefficient is != 0?
Null hypotheses would generally be null - either 'no effect' or some conventionally accepted value. In this case, the population coefficient being 0 is a classical 'no effect' null.
More prosaically, when testing a point hypothesis against a composite alternative (a two-sided alternative in this case), one takes the point hypothesis as the null, because that's the one under which we can compute the distribution of the test statistic (more generally, using an open set for a null presents certain problems, even when both are composite). With a pair of point hypotheses, one is (at least mechanically) free to make either one the null (and even then one still would generally want to make the one that's most clearly "null" the null -- if either of them is; that is to choose the 'no effect' or conventionally-accepted one the null).