Since stress is usually the highest on the surface, most of the time you should be looking for the values on the surface = nodes.

Now since FE calculates stresses at integration points, then extrapolates it to the nodes, you have to make sure, that the element size/formulation is appropriate to capture the stresses you are looking for. In the end, a correctly meshed and converged solution needs to give very close answers. The larger the element and the stress gradient, the bigger difference you will see. If the stress you want to capture changes significantly based on which element/where you measure you need to refine until you get convergence.

I think it depends on your interest. there is no much difference between the two methods, I usually focus on nodal solution because I apply the load on nodes so I need to check the results on each node.

Good question.. Please share me the best answer might you get...

In ANSYS the default is Nodal stresses. Nodal stress and element stress are two ways of representing the same data.

As other answers have mentioned that the values are calculated at locations called integration/Gaussian points. These values are then extrapolated to nodes. If the averaging is done at the node, the stresses obtained are called Nodal stresses. If done for an element the stresses are called elemental stresses. For a refined mesh the two values should be similar to each other. Also, it is a good practice to check element stress since it will give an idea as to which region needs refinement. The more refined the mesh is smoother is the transition.

Doesn't that depend on the application?

for example, I'm interested in elemental failure. My postprocessing requires that I decide whether an (element) failed or not based on their stress or strain (depending on the failure criterion I'm using). This means I need to use elemental values.

As Claudio Pedrazzi points out an old school way of checking if the mesh is adequate (refined enough) is by comparing nodal and elemental stresses. They should be fairly closed unless your mesh is coarse or you're dealing with a singularity. Also, it is worth pointing that indeed you should keep an eye on the failure modes of the material you are using. Take as an example Brittle Materials. Some of them possess a large critical distance (if you're not familiar with the topic search for Susmel and David Taylor), meaning that you shouldn't be looking for the highest stresses on the surface but rather at a certain distance (L/2) from the surface. This theory is based on the fact that certain cracks may nucleate at the surface but stop at half the critical distance. In other words, if you can live with non-propagating cracks, you'd be better off looking at the stresses happening at the critical distance, as you can claim some designing advantage.