We are used to doing our modeling with quads, meaning polygons that have exactly 4 sides. This, however, is only an abstraction and what we actually have are two triangles that are contained in this 4-sided polygon. All quads (and generally speaking, polygons with more than 3 sides) are always made up of triangles. You may not see it but your offline renderer, the viewport in your 3d application, and game engines all only use triangles.
A 4x4 quads plane on the left and the hidden triangulation made visible on the right
Quads and consequently all larger n-gons (polygons with n sides) only exist as a modeling aid. It's more convenient to subdivide a quad than subdividing triangles where the direction of the flow of the mesh is not obvious. N-gons are also useful for capping flat holes where the exact way the polygon is triangulated doesn't necessarily matter. Using n-gons, however, is rather frowned upon and usually for good reasons. One reason is that the automatic triangulation can be very irregular and produce issues in certain circumstances.
All right, so far so good, but if we prefer to use quads anyways, why should we care about the underlying triangulation?
Let's look at a few examples and how the triangulation can affect our results.
When it suddenly matters
different triangulation direction
It not usually a problem with high-poly meshes and less extreme curvatures but it is much more likely to be a problem in game models, especially for mobile applications where the triangle budget is much more limited. So you should be aware that automatic triangulation can produce unintended results.
overlapping faces due to triangulation
In cases like this, it is advisable to add edges manually to prevent triangulation in concave shapes. In this example, you could simply connect the left corners to the corners in the middle and you would end up with quads again.
Another important case where triangulation matters is the export and import into other 3D applications combined with baked normals. The automatic triangulation between your content creation program and, for example, your game engine, will likely differ and suddenly you will end up with ugly normal map artifacts. This is due to resulting tangent space mismatches. The tangent space is calculated using those hidden triangles, so if those triangles change between your applications, the tangent space will change too. See the "Tangent space" chapter for a deeper explanation. This one of the reasons why you usually want to triangulate your model before baking and export