How do atoms make colours?

The biggest effect is actually what colour something absorbs...

Different colours of light have different energies. The bluer the light, and light comes in blobs called photons, the more energy the photon has. 

The electrons inside atoms can only have certain energies so they have what are called energy levels. Maybe they can absorb a certain amount of energy or twice that amount or two and a half times that amount, changing in discrete amounts. They move up and down energy 'shells', between orbitals and energy levels.

A substance can only absorb light if the difference between one energy level and the next is equal to the energy of the photon. It will gain the right amount of energy to absorb that photon of light. This is also is why substances have that characteristic spectral fingerprint, an absorption pattern of certain frequencies of light that they absorb and certain lights that they reflect.

Fluorescence is a related effect. This happens when you get high energy photons, for example ultraviolet light, which hit an atom. The atom will absorb that energy and then instead of releasing it all in one big lump, it releases it in two or three smaller lumps which will be a different colour to the UV, a lower frequency that you can see. So something can absorb ultraviolet light, then emit blue light or green light and it looks like it's glowing. 

Another thing a lot of people don't often realise is that most of the different colours in nature are flowers. And these colours are actually created by the same molecule, a molecule called anthocyanine. What the flower does is to make the petal more acidic or alkaline. What this does is to add or remove hydrogen from the molecule and this changes the way the electrons whizz around in the molecule. This, in turn, changes the light wavelengths that they can soak up. The petals may be basically different colours, but it's all down to the same chemical, anthocyanine!