The characteristics are the lines y=a (a an arbitrary constant). Here's a plot of this surface (z=u(x,y)) on the coordinate axes (or dowload it as an eps file). Some of the characteristic lines are drawn in blue in the x-y plane. (Notice the labeled x- and y-axes are actually pointing in the -x and -y directions.)

MATLAB code to draw this:
    >> hold off;
    >> [X,Y]=meshgrid(-5:.01:5,-5:.01:5);
    >> mesh(X,Y,sin(Y));
    >> hold on;
    >> axis([-5,5,-5,5,-1.3,1.3]);
    >> for i = -5:5
    x=-5:.01:5;
    y=i*ones(size(x))
    plot3(x,y,-1.3*ones(size(x)));
    end

Example 2: The PDE u_x-3u_y = 0 with the condition u(x,x)=x^2 has the unique solution

This surface is plotted here (eps file here), along with some of the characteristics 3x+y=a. Notice that u is constant along each characteristic, and also that along the line y=x (shown in red in the x-y plane), u(x,x)=x^2 (see the black line shown on the surface).

MATLAB code:
    >> hold off;
    >> [X,Y]=meshgrid(-2:.01:2,-2:.01:2);
    >> mesh(X,Y,(3*X+Y).^2/16);
    >> hold on;
    >> axis([-2,2,-2,2,-.2,4.2]);
    >> for i=-7:7
    x1=max(-2,(i-2)/3);
    x2=min(2,(i+2)/3);
    x=x1:.01:x2;
    y=-3*x+i;
    plot3(x,y,-.2*ones(size(x)),'blue');
    end
    >> x=-2:.01:2;
    >> plot3(x,x,zeros(size(x)),'red');
    >> plot3(x,x,x.^2,'black');

Example 3: When we solve u_x + (xy)u_y = 0 along with the condition u(0,y)=exp(-y^2), we find the unique solution

The surface z=u(x,y) is plotted here (or as an eps file). The characteristics are shown in the x-y plane in blue; notice that u is constant along the characteristics and also that along the y-axis, u satisfies the initial condition u(0,y)=exp(-y^2).

MATLAB code:
    >> hold off;
    >> [X,Y]=meshgrid(-2:.01:2,-2:.01:2);
    >> mesh(X,Y,exp(-Y.^2.*exp(-X.^2)));
    >> hold on;
    >> axis([-2,2,-2,2,-.1,1.1]);
    >> for i=[.25,.5,1,1.5]
    x1=max(-2,-sqrt(2*log(2/i)));
    x2=min(2,sqrt(2*log(2/i)));
    x=x1:.01:x2;
    plot3(x,i*exp(x.^2/2),-.1*ones(size(x)),'blue');
    plot3(x,-i*exp(x.^2/2),-.1*ones(size(x)),'blue');
    end
    >> y=-2:.01:2;
    >> plot3(zeros(size(y)),y,exp(-y.^2),'black');