What is the difference between 1-dim.harmonic oscillator and 2-dim. harmonic oscillator? I ask myself what exactly is meant with "2-dimensional harmonic oscillator".
I only know the situation of a bob hanging on a bar... is that 1-dimensional or 2-dimensional?

The difference is the number of spatial dimensions in which the oscillator is allowed to oscillate. 
The 1D oscillator has a potential function,
$$ V(x) = kx^2, $$
where the 2D oscillator has the potential function,
$$ V(x,y) = k (x^2 + y^2) .$$
In the context of classical mechanics the differential equations for $x$ and $y$ can be obtained by,
$$ \frac{d^2x}{dt^2} = -\frac{1}{m}\frac{\partial V}{\partial x} = -\frac{k}{m}x, $$
$$ \frac{d^2y}{dt^2} = -\frac{1}{m}\frac{\partial V}{\partial y} = -\frac{k}{m}y. $$
The solutions are, $$x(t) = A \sin\left(\sqrt{\frac{k}{m}}t + \delta \right),$$ $$y(t) = B \sin\left(\sqrt{\frac{k}{m}}t + \delta' \right).$$

For a concrete example consider a bead at the bottom of a bowl. For small deflections from the minimum of the bowl the bead will oscillate harmonically in two dimensions. 
