Getting smooth, big points in OpenGL

Question

I started playing around with OpenGL and GLUT. I would like to draw some points, but the problem is that they turn out to be squares, and I would like them to be round dots (filled circles).

This is what I do:

void onInitialization( ) 
{ 
    glEnable( GL_POINT_SMOOTH );
    glEnable( GL_BLEND );
    glBlendFunc( GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA );
    glPointSize( 6.0 );
}    

void onDisplay()
{
    glClearColor( 1.0f, 1.0f, 1.0f, 1.0f );
    glClear( GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT );

    glBegin( GL_POINTS );
        glColor3f( 0.95f, 0.207, 0.031f );
    for ( int i = 0; i < g_numPoints; ++i )
    {
        glVertex2f( g_points[i].X, g_points[i].Y );
    }
    glEnd();
    glFinish();
    glutSwapBuffers();
}

This is the result:

The points show up where expected, only their shape is wrong.

Solution

Mads' answer provides everything you need if you go for the fixed function pipeline. However, if you have a system that does not provide the ARB_point_sprite extension or with a broken implementation (some ATI drivers), you can solve this part also with geometry shaders. The ARB_geometry_shader4 extension allows you to convert a point primitive to two triangles, which can be used as the quad created by the ARB_point_sprite extension. On OpenGL 3.2, geometry shaders are already supported in core, no extension needed. The OpenGL wiki has two examples.