in front of the one-dimensional Gaussian kernel is the normalization constant. It comes from the fact that the integral over the exponential function is not unity: ¾- e- x2 2 s 2 Ç x = !!!!! !!! 2 p s . With the normalization constant this Gaussian kernel is a normalized kernel, i.e. its integral over its full domain is unity for every s .
The linear kernel is what you would expect, a linear model. I believe that the polynomial kernel is similar, but the boundary is of some defined but arbitrary order (e.g. order 3: $ a= b_1 + b_2 \cdot X + b_3 \cdot X^2 + b_4 \cdot X^3$).