When doing least squares estimation (assuming a normal random component) the regression parameter estimates are normally distributed with mean equal to the true regression parameter and covariance matrix Σ=s2⋅(XTX)−1 where s2 is the residual variance and XTX is the design matrix. XT is the transpose of X and X is defined by the model equation Y=Xβ+ϵ with β the regression parameters and ϵ is the error term. The estimated standard deviation of a beta parameter is gotten by taking the corresponding term in (XTX)−1 multiplying it by the sample estimate of the residual variance and then taking the square root. This is not a very simple calculation but any software package will compute it for you and provide it in the output.
Example
On page 134 of Draper and Smith (referenced in my comment), they provide the following data for fitting by least squares a model Y=β0+β1X+ε where ε∼N(0,Iσ2).
X Y XY
0 -2 0
2 0 0
2 2 4
5 1 5
5 3 15
9 1 9
9 0 0
9 0 0
9 1 9
10 -1 -10
--- -- ---
Sum 60 5 32
Sum of Squares 482 21 528
Looks like an example where the slope should be close to 0.
Xt=(101212151519191919110).
So
XtX=(n∑Xi∑Xi∑X2i)=(106060482)
and
(XtX)−1=⎛⎝⎜⎜∑X2in∑(Xi−X¯)2−X¯∑(Xi−X¯)2−X¯∑(Xi−X¯)21∑(Xi−X¯)2⎞⎠⎟⎟=(48210(122)−6122−61221122)=(0.395−0.049−0.0490.008)
where X¯=∑Xi/n=60/10=6.
Estimate for β=(XTX)−1XTY = ( b0 ) =(Yb-b1 Xb)
b1 Sxy/Sxx
b1 = 1/61 = 0.0163 and b0 = 0.5- 0.0163(6) = 0.402
From (XTX)−1 above Sb1 =Se (0.008) and Sb0=Se(0.395) where Se is the estimated standard deviation for the error term. Se =√2.3085.
Sorry that the equations didn't carry subscripting and superscripting when I cut and pasted them. The table didn't reproduce well either because the spaces got ignored. The first string of 3 numbers correspond to the first values of X Y and XY and the same for the followinf strings of three. After Sum comes the sums for X Y and XY respectively and then the sum of squares for X Y and XY respectively. The 2x2 matrices got messed up too. The values after the brackets should be in brackets underneath the numbers to the left.