Reductionism is the idea that all known phenomena are the simple sum of simpler, more fundamental ones.
It has worked really well in science, and it could even be argued that reduction is a good part of what science is.
Chemistry is well explained by the interactions of atoms, atoms are well understood as a quantum assemblage of protons, neutrons and electrons, and protons and neutrons are quarks bound by their strong interaction mediated by gluons.
Another example of very successful reduction is thermodynamics. Thermodynamics provide a set of consistent and successful laws that rule the behavior of macroscopic quantities such as temperature, pressure or volume. You do not need to understand the microscopic dynamics of atoms and molecules in a gas to use it and understand how a fridge or a thermal engine work. Still it’s true that using only statistics and some simple dynamics, you can derive all the laws of thermodynamics. Reductionism win!
Now take higher-level phenomena such as sociology or psychology. It would seem absurd to claim that these can be reduced to the quantum interaction of subatomic particles. More importantly, it would be sterile and counter-productive to attempt it. Not to mention impossible in practice.
A second way in which reductionism can fail besides practical irrelevance is in the assumption that you can effectively separate reality in distinct layers that don’t interact with one another. In particular with non-linear or chaotic phenomena, small uncertainties in the state of lower layers can translate into very large differences in the higher layers (the so-called and much misunderstood butterfly effect). This also means that in turn, higher layers can affect the lower ones in inextricable ways.
Take evolution for example. Biology is reducible in principle to chemistry (and biochemistry is a triumphant discipline that has probably done more for the betterment of humankind than most), but the interactions of living bodies run so deep and are so tied to environmental factors (even though those are also reducible in principle), and they do in turn affect their own environment in such important ways that it is impossible to give a complete picture of evolution based only on chemistry.
But in principle, each layer does strictly depend on the underlying, more fundamental layer. To this day, there are no phenomena that expose demonstrable contradictions with lower layers. Such a contradiction could come for example in the form of a macroscopic phenomenon not conserving energy. Here you would have a quantity that is valid at both levels but that would behave differently at the higher level than it does at the lower one. This never happened so far.
In other words, the failures of reductionism are not failures of the principle, they are failures of practical applicability and relevance.
Reductionism is not sufficient anymore in the scientific arsenal. It does however remain extremely useful and we do still rely in many cases on its theoretical validity.