Why don't insulators conduct electricity? I have been taught in school that because of the high energy band gap (approx. $9~\text{eV}$ or more) of an insulator, electrons can't jump to the conduction band.
But also $1~\text{eV} = 1.6 \cdot 10^{-19}~\text{J}$ which is much less. So why can't an electron jump to the conduction band when it requires such a low energy in the case of the insulator?
 A: If you have a chunk of insulator sitting on your workbench at a temperature $T$ then the energy available for exciting electrons is of order $kT$, where $k$ is Boltzmann's constant.
If we take room temperature, say $T=300\textrm{K}$, then $kT \approx 4.4 \times 10^{-21}$ or about $0.025\textrm{eV}$. And this is why electrons can't be excited into the conduction band. You mention a band gap of $9\textrm{eV}$ and since the thermal energy available is only $0.025\textrm{eV}$ there isn't enough energy to cross the band gap.
You say that $1\textrm{eV} = 1.602 \times 10^{-19}$ joules is a small amount of energy, and yes it is. But it's still much large than the energy available at room temperature.
If you provide more energy then you can turn your insulator into a (semi)conductor. For example if you shine hard ultraviolet light with a photon energy of $9\textrm{eV}$ or greater onto your insulator it will excite electrons into the conduction band and as a result the insulator will start to conduct electricity.
A: 
But also $1~\text{eV} = 1.6 \cdot 10^{-19}~\text{J}$ which is much less.

That "$=$" is saying that no, the right hand side is not much less than the left hand side. Both are the same, 1 electron Volt is the same amount of energy as $1.6 \cdot 10^{-19}$ Joule.
This energy is of course much smaller than anything you might usually get in mechanics.
It might help to think about the band gap energy as "energy needed for each electron" to get to the conduction band. Statistically, there will be very few electrons who might make the jump, but it is very unlikely that a huge number of electrons get this tiny amount of energy from somewhere to get to the conduction band.
There are ways to get the electrons of an insulator to the conduction band, for example with appropriate light (see John Rennie's answer) or by heating a glass rod until it almost melts. It will become somewhat conducting.
