a1 is an amplitude parameter defining the height of the guassian.
b1 is the "mu" parameter defining the center.
c1 is the "sigma" parameter defining its width.
This is a typical form of the gaussian function though it can take other forms and other parameters. The first function you shared is in the form of a probability density function (which is in the guassian family). Wiki does a good job of describing the difference.
Plot them to show their differences.
f1 = @(x, mu, sigma) (1./sigma.*sqrt(2*pi)).* exp((-1/2)*((x-mu)./sigma).^2);
f2 = @(x,a1,b1,c1) a1.*exp(-((x-b1)./c1).^2);
x = linspace(0,1,100);
mu = .4;
sigma = .2;
amp = 1./sigma.*sqrt(2*pi); % same amp as f1
y1 = f1(x,mu,sigma);
y2 = f2(x,amp,mu,sigma);
cla()
hold on
plot(x,y1, 'b-', 'linewidth', 2,'DisplayName', 'f1');
plot(x,y2, 'r-', 'linewidth',2,'DisplayName', 'f2');
legend
Higher order guassians (aka super-gaussian) functions have a parameter that flattens the top which gradually leaves the guassian family (example).
