What is hylomophism and how can it be related with matter and form, being and existence
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Hylomorphism is the view propounded by Aristotle where things are understood as being formed matter. This is implicit in the Greek words in the term hylomorphism with hyle being matter and morphe being form.
In the case of living things, these forms are self-managing and arranging and are called "souls". For Aristotle, plants, animals, and humans all have souls but souls that differ in their functions. All of the souls engage in reproduction and growth but animals add to this sensation and humans reason.
I am not quite sure what the phrase "Hylomorphism as the foundation of matter" would mean.
But Aristotle did believe the only sorts of things we encounter are formed matter. Part of this is due to a difficult point in his metaphysics that plays on the pair potential and actual. Matter is fundamentally potential in that it can be shaped and organized. Form conversely is fundamentally active as it organizes and arranges something in a particular way.
(A further complexity is the way artifacts (which for Aristotle means things people make) work insofar as the artifact's ideaa is not an intrinsic form, -- e.g., there's no essence he calls table).
The key distinctive between this and Plato's view is that for Aristotle we observe forms using the essences and then abstract these from things. Similarly, the world we live in is not a world of shadow for Aristotle but the source of what we know.
Hylomorphism is the name of one of Aristotle's basic principles to investigate the material world. The name hylomorphism is explained in virmaior's answer. The term itself is not due to Aristotle, it has been coined much later.
Aristotle introduces the principle of hylomorphism in Metaphysics, book I, chapter 3. In this book Aristotle presents the first European history of philosophy. He recalls and classifies the philosophy of nature as given by his forerunners, the Ionian and Italian philosophers as well as his teacher Plato. Aristotle presents theirs views, gives a critical evaluation and compares them to his own principle of explanation.
Aristotle's principle is the doctrine of the four causes, in Latin: causa materialis, causa formalis, causa efficiens and causa finalis (Chapt. 7).
- Hyle refers to causa materialis: What is the thing made from?
- Morphe refers to causa formalis: Which form has the thing? The question relates to what Plato calls the idea.
- Causa efficiens asks: What is the cause of its existence? That is today's principle of causality.
- Causa finalis: To which goal does the thing move or develop?
Aristotle states that asking these four questions is more powerful and prompts a more comprehensive answer than the partial aspects considered by his forerunners.
In my opinion, Aristotle is right in this historical assessment. But a modern point of view criticizes that it is not meaningful to pose all questions for every object, e.g. it is not meaningful to ask for the causa finalis of inanimate objects.
A second point refers to the scope of his principle. The scope, where Aristotle's principle can be usefully applied, covers only objects from our everyday life, the mesocosm. Science like physics has expanded the domain of investigation to microcosm and macrocosm. E.g. the guiding principle from particle physics is to investigate the different kinds of interaction forces. And to study the possible ways to create particles from pure (= non-material) energy and to annihilate particles into pure energy.