The net effect for forming an ionic bond between two atoms is the sum of all of the "parts" of making the two ions from the elements and the energy of bringing them together. However, we can understand a lot about ion compounds simply by thinking about the interactions of the two ions (assuming they were already formed somewhere in the world).
The potential energy of two charges interacting is given by Coulomb's Law. This states that energy is related to the charges and the distance between them where
\[E \propto { {q_1q_2} \over \epsilon \;r} \]
where \(q_1\) and \(q_2\) are the charges on the ions and \(r\) is the distance between them. \(\epsilon \) is the dielectric constant which for vacuum (gas phase) is simply equal to one. Thus if one charge is positive and the other is negative the energy of this interaction will be negative. That is to say, the two ions will have a lower energy (be more stable) when they are close together than when they are far apart. The magnitude of this energy depends on the size of the charges and the distances between them. Larger charges will have stronger interactions and lower energies. Likewise smaller ions will be able to get closer together and thus will have lower energies. Sometimes we take these two ideas and combine them into one we call charge density. How much charge is in a space. The larger the charge the larger the charge density. The smaller the ion the larger the charge density.
For example, the lattice energy of LiF will be higher than that of KF. Since Li+ is smaller (has a higher charge density) than K+, the ions in LiF are at a closer separation than the ions in KF. Similarly the lattice energy of NaF is higher than NaBr since, F- is smaller than Br-.
Coulomb's Law looks at the interaction of only two ions. In ionic compounds, we have a lattice that is composed of many ions all of which are interacting. The energy that is released upon forming the lattice from separate ions is the called the lattice energy. This essentially is the difference in energy for the separated ions compared to the ions in the crystal lattice. The lattice energy is complicated as it depends on the structure of the crystal. However, in general if we have similar crystal structures, then the differences between different ions will be given by the difference in the Coulomb energy. So we can approximate the magnitude of the lattice energy as simply being proportional to the Coulomb energy.
The lattice energy is defined as the energy required to separate the crystalline solid into gas phase ions as is shown in the example below for KCl.
\[ {\rm KCl(s) \rightarrow K^+(g) + Cl^-(g)}\]
For KCl the lattice energy is 715 kJ mol-1. This is a lot of energy. As a result chemists will often say that ionic bonds are very strong. This is because they are referring to this particular situation of taking the solid and turning it into gas phase ions.
In contrast, you will often hear in biology that ionic bonds are the weakest bonds! This because they are generally talking about chemistry in a very specific context: aqueous solutions.
\[ {\rm KCl(s) \rightarrow K^+(aq) + Cl^-(aq)}\]
Chemistry is water is very different. Now if you are looking to break apart ions they are not ending up in the gas phase but instead they are "aqueous" or surround by water molecules denoted by the (aq) in the equation. This stabilizes their energy making it much less difficult to break apart the ionic solid. Additionally, the interactions between the ions is not in a vacuum but instead takes place in water. The dielectric constant, \(\epsilon\), for water is about 80. This means the attractive force and potential energy are effectively 80 times lower in water than in the gas phase.