The presocratic philosophers are commonly understood to be those thinkers who lived before Socrates (hence the name). However, that is not necessarily the case,as some of the presocratics were contemporaries of Socrates, and one may have been his teacher (Archelaus). These philosophers were concerned for the most part with nature, and the ultimate underlying principle or element (the arche).
Here is a general guide on the presocratic philosophers from the Stanford Encyclopedia of Philosophy: Presocratic Philosophy
"Thales was interested in almost everything, investigating almost all areas of knowledge, philosophy, history, science, mathematics, engineering, geography, and politics. He proposed theories to explain many of the events of nature, the primary substance, the support of the earth, and the cause of change. Thales was much involved in the problems of astronomy and provided a number of explanations of cosmological events which traditionally involved supernatural entities. His questioning approach to the understanding of heavenly phenomena was the beginning of Greek astronomy. Thales’ hypotheses were new and bold, and in freeing phenomena from godly intervention, he paved the way towards scientific endeavor. He founded the Milesian school of natural philosophy, developed the scientific method, and initiated the first western enlightenment. A number of anecdotes is closely connected to Thales’ investigations of the cosmos. When considered in association with his hypotheses they take on added meaning and are most enlightening. Thales was highly esteemed in ancient times, and a letter cited by Diogenes Laertius, and purporting to be from Anaximenes to Pythagoras, advised that all our discourse should begin with a reference to Thales (D.L. II.4)."
"His work will always remain truncated, like the mutilated and decapitated statue that has been found at the market-place of Miletus and that bears his name. Nevertheless, by what we know of him, we may say that he was one of the greatest minds that ever lived. By speculating and arguing about the “Boundless” he was the first metaphysician. By drawing a map of the world he was the first geographer. But above all, by boldly speculating about the universe he broke with the ancient image of the celestial vault and became the discoverer of the Western world-picture."
"Anaximenes is best known for his doctrine that air is the source of all things. In this way, he differed with his predecessors like Thales, who held that water is the source of all things, and Anaximander, who thought that all things came from an unspecified boundless stuff."
"As the first philosopher to inquire into the nature of existence itself, he is incontrovertibly credited as the “Father of Metaphysics.” As the first to employ deductive, a priori arguments to justify his claims, he competes with Aristotle for the title “Father of Logic.” He is also commonly thought of as the founder of the “Eleatic School” of thought—a philosophical label ascribed to Presocratics who purportedly argued that reality is in some sense a unified and unchanging singular entity. This has often been understood to mean there is just one thing in all of existence. In light of this questionable interpretation, Parmenides has traditionally been viewed as a pivotal figure in the history of philosophy: one who challenged the physical systems of his predecessors and set forth for his successors the metaphysical criteria any successful system must meet."
Zeno of Elea
"In the fifth century B.C.E., Zeno of Elea offered arguments that led to conclusions contradicting what we all know from our physical experience—that runners run, that arrows fly, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because many of the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno was the first person to show that the concept of infinity is problematical. In the Achilles Paradox, Achilles races to catch a slower runner—for example, a tortoise that is crawling in a line away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least as far as the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run at least to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones."
"Melissus was a follower of Parmenides’ thought but not in all its details. On the one hand, Melissus agrees with Parmenides’ main arguments on the indestructibility, immobility, indivisibility, oneness, completeness, changelessness and perfection of Being. On the other, he adopts a different viewpoint on the Parmenidean timelessness and finitude of Being. Melissus understood non-being in terms of spatial emptiness. Since non-being is impossible as an enclosing limit, then Being is limitless. Thus, while Parmenides’ Being is timeless in finitude, Melissus’ Being is everlasting in infinitum."
"Xenophanes of Colophon was a philosophically-minded poet who lived in various parts of the ancient Greek world during the late 6th and early 5th centuries BCE He is best remembered for a novel critique of anthropomorphism in religion, a partial advance toward monotheism, and some pioneering reflections on the conditions of knowledge. Many later writers, perhaps influenced by two brief characterizations of Xenophanes by Plato (Sophist 242c–d) and Aristotle (Metaphysics 986b18-27), identified him as the founder of Eleatic philosophy (the view that, despite appearances, what there is is a changeless, motionless, and eternal ‘One’). In fact, the Xenophanes who emerges from the surviving fragments defies simple classification. He was a travelling rhapsode who criticised the stories about the gods told by the poets, and he defended a novel conception of the divine nature. But he was also a reflective observer of the human condition, a practitioner of the special form of ‘inquiry’ (historiê) introduced by the Milesian philosopher-scientists, and a civic counselor who encouraged his fellow citizens to respect the gods and work to safeguard the well-being of their city."
"He claims to announce an everlasting Word (Logos) according to which all things are one, in some sense. Opposites are necessary for life, but they are unified in a system of balanced exchanges. The world itself consists of a law-like interchange of elements, symbolized by fire. Thus the world is not to be identified with any particular substance, but rather with an ongoing process governed by a law of change. The underlying law of nature also manifests itself as a moral law for human beings. Heraclitus is the first Western philosopher to go beyond physical theory in search of metaphysical foundations and moral applications."
"The pre-Socratic Greek philosopher Pythagoras must have been one of the world’s greatest persons, but he wrote nothing, and it is hard to say how much of the doctrine we know as Pythagorean is due to the founder of the society and how much is later development. It is also hard to say how much of what we are told about the life of Pythagoras is trustworthy; for a mass of legend gathered around his name at an early date. Sometimes he is represented as a man of science, and sometimes as a preacher of mystic doctrines, and we might be tempted to regard one or other of those characters as alone historical. The truth is that there is no need to reject either of the traditional views. The union of mathematical genius and mysticism is common enough. Originally from Samos, Pythagoras founded at Kroton (in southern Italy) a society which was at once a religious community and a scientific school. Such a body was bound to excite jealousy and mistrust, and we hear of many struggles. Pythagoras himself had to flee from Kroton to Metapontion, where he died."
If you want to know more about Pythagorean philosophy, see: Pythagoreanism
"The sophists were itinerant professional teachers and intellectuals who frequented Athens and other Greek cities in the second half of the fifth century B.C.E. In return for a fee, the sophists offered young wealthy Greek men an education in aretē (virtue or excellence), thereby attaining wealth and fame while also arousing significant antipathy. Prior to the fifth century B.C.E., aretē was predominately associated with aristocratic warrior virtues such as courage and physical strength. In democratic Athens of the latter fifth century B.C.E., however, aretē was increasingly understood in terms of the ability to influence one’s fellow citizens in political gatherings through rhetorical persuasion; the sophistic education both grew out of and exploited this shift. The most famous representatives of the sophistic movement are Protagoras, Gorgias, Antiphon, Hippias, Prodicus and Thrasymachus."
"A number of important theorists in ancient Greek natural philosophy held that the universe is composed of physical ‘atoms’, literally ‘uncuttables’. Some of these figures are treated in more depth in other articles in this encyclopedia: the reader is encouraged to consult individual entries on Leucippus, Democritus, Epicurus and Lucretius. These philosophers developed a systematic and comprehensive natural philosophy accounting for the origins of everything from the interaction of indivisible bodies, as these atoms—which have only a few intrinsic properties like size and shape—strike against one another, rebound and interlock in an infinite void. This atomist natural philosophy eschewed teleological explanation and denied divine intervention or design, regarding every composite of atoms as produced purely by material interactions of bodies, and accounting for the perceived properties of macroscopic bodies as produced by these same atomic interactions. Atomists formulated views on ethics, theology, political philosophy and epistemology consistent with this physical system. This powerful and consistent materialism, somewhat modified from its original form by Epicurus, was regarded by Aristotle as a chief competitor to teleological natural philosophy."
"Democritus, known in antiquity as the ‘laughing philosopher’ because of his emphasis on the value of ‘cheerfulness,’ was one of the two founders of ancient atomist theory. He elaborated a system originated by his teacher Leucippus into a materialist account of the natural world. The atomists held that there are smallest indivisible bodies from which everything else is composed, and that these move about in an infinite void. Of the ancient materialist accounts of the natural world which did not rely on some kind of teleology or purpose to account for the apparent order and regularity found in the world, atomism was the most influential. Even its chief critic, Aristotle, praised Democritus for arguing from sound considerations appropriate to natural philosophy."
"His works On Nature and Purifications (whether they are two poems or only one – see below) exist in more than 150 fragments. He has been regarded variously as a materialist physicist, a shamanic magician, a mystical theologian, a healer, a democratic politician, a living god, and a fraud. To him is attributed the invention of the four-element theory of matter (earth, air, fire, and water), one of the earliest theories of particle physics, put forward seemingly to rescue the phenomenal world from the static monism of Parmenides."
"Anaxagoras of Clazomenae was an important Presocratic natural philosopher and scientist who lived and taught in Athens for approximately thirty years. He gained notoriety for his materialistic views, particularly his contention that the sun was a fiery rock. This led to charges of impiety, and he was sentenced to death by the Athenian court. He avoided this penalty by leaving Athens, and he spent his remaining years in exile. While Anaxagoras proposed theories on a variety of subjects, he is most noted for two theories. First, he speculated that in the physical world everything contains a portion of everything else. His observation of how nutrition works in animals led him to conclude that in order for the food an animal eats to turn into bone, hair, flesh, and so forth, it must already contain all of those constituents within it. The second theory of significance is Anaxagoras’ postulation of Mind (Nous) as the initiating and governing principle of the cosmos."