What is the difference between an element and an atom?
The wording in your quote is a bit unclear, I agree. A very nice definition is given by the Jefferson Lab page What is the simplest way of explaining what atoms, elements, compounds and mixtures are?

Atoms


  are the smallest bits of ordinary matter


Whereas an element is


  a substance that is made entirely from one type of atom


Atoms themselves can be 'broken down' into smaller sub-atomic particles (protons, neutrons and electrons) - the amounts of these (particularly the number of protons) define each element.

For example, in their un-ionised form, the element carbon is made entirely from carbon atoms, which have 6 protons and 6 electrons each, and cadmium atoms, which have 48 protons and 48 electrons, make up the element cadmium.

The amount of neutrons can vary with atoms of a particular element, forming isotopes, which (from the link):


  For example, carbon-12, carbon-13 and carbon-14 are three isotopes of the element carbon with mass numbers 12, 13 and 14 respectively. The atomic number of carbon is 6, which means that every carbon atom has 6 protons, so that the neutron numbers of these isotopes are 6, 7 and 8 respectively.


Different arrangements of atoms of single elements can form allotropes, such as for carbon, with diamond and graphite (among others - shown below). They both are composed of atoms of the element carbon, but in different configurations.

Below are diagrams of the allotropes of carbon featured in the article Work Function Engineering of Graphene (Garg et al. 2014)