In the first case, $x(0)$ and $x(1)$ and $x(2)$ etc are all uncorrelated. (And since they're gaussian, they're independent). This means that, to get a realization of the first random process, you have to take $\infty$ independent samples from a gaussian distribution, and assign each of these $\infty$ samples to a $x(n)$.
The second process is much simpler. You just take one sample from a gaussian distribution (and call it $\alpha$), and then assign all of the $x(n)$ to this $\alpha$. In this way, $\alpha$ may be random, but you will get $x(0)=x(1)=x(2)=\cdots=\alpha$.
I'll post some plots, as you asked. I used the following code in MATLAB:
N = 3;
T = 10;
t = 0:T;
x1 = randn(N,T+1);
x2 = repmat(randn(N,1), 1, T+1);
subplot(3,1,1); stem(t, x1(1,:));
subplot(3,1,2); stem(t, x1(2,:));
subplot(3,1,3); stem(t, x1(3,:));
subplot(3,1,1); stem(t, x2(1,:));
subplot(3,1,2); stem(t, x2(2,:));
subplot(3,1,3); stem(t, x2(3,:));
In this first figure, I plot three independent realizations of Random Process 1:
And in this figure, three independent realizations of Random Process 2. Note that in this case, all the $x$ signals are constant, but the actual level changes from one realization to another: