Peter Flom answer is incorrect. I don't have the reputation to comment on Peter Flom question, so I'll place my response here.
Let's use the example of a factor of color that can be Red, Green, Blue. Let's pretend that these colors will correspond to an average response variable (y) of 40, 60, and 30 respectively.
Now it is not commonly thought about, but ANOVA and linear regression are the actually exactly the same thing. The design matrix (the X) of the Linear model y = Xb + e would look something like this...
R G B
1 1 0 0
1 0 1 0
1 0 0 1
...however when attempting to estimating the coefficients by the derivative of the sum of square errors with respect to b (that is b = (X^T * X) ^-1 * X^T * y you will notice that X^T * X is a singular matrix. If you think about it, this makes intuitive sense. The work around for this is simple. You turn one treatment into the intercept and express all avg responses in relation to that intercept. See new design matrix below...
(R) G B
1 0 0
1 1 0
1 0 1
...now we have a design matrix where the intercept is actually the treatment Red. All average responses are now with respect to RED ie. Red = 40, Green = 20, and Blue = -10.
R = R = 40
G = G + R = 20 + 40 = 60
B = B + R = -10 + 40 = 30
In other words in an ANOVA (which is really the same as a linear regression) the intercept is actually a treatment and a significant intercept means that treatment is significant. Now if you get into two way or even higher levels of ANOVA the interpretation of the intercept becomes more complex, but for a one way anova the intercept is itself a just another treatment.