Warning: Unmarked spoilers follow.
It seems there are four different components you're addressing here, so I'll make the differences for each one clear.
Van Hohenheim, also known as slave #23, was a citizen of Xerxes and a regular human being who was employed by King Xerxes' alchemist. This alchemist was an experimenter, and managed to create a being, the Dwarf in the Flask, as a result of alchemy.
The Dwarf gave the slave a name, Van Hohenheim, before allying with him in order to turn the citizens of Xerxes into a philosopher's stone. Once this was successful, Hohenheim, as a human, had over 500,000 souls within his blood, giving him great power. So, while he is a real human being, he is also a philosopher's stone.
Homunculus (in the Flask)
Homunculus, better known as Dwarf in the Flask, was created by the King's alchemist (as briefly mentioned above). This alchemist (who is not named) was experimenting with the Gate, and brought forth some of the essence of "God" to create the Dwarf in the Flask, the first true homunculus. He has no physical form, and can only reside within the glass flask he was created in.
Father is the result of mixing the Dwarf with the Xerxes transmutation circle: The Dwarf in the Flask used some of the power from the resulting Xerxes transmutation to give himself a false human body (based on Hohenheim's). As Hohenheim states, Father started out as nothing more than the same Dwarf, contained inside a human "flask" (rather than a glass one).
As you see in the series, he later evolves quite substantially; he is the one who swallows God, and becomes the ultimate being. However, he is not a real human; he remains a homunculus (a manufactured human).
The homunculi are beings (false humans) created from Father's blood, which is essentially a philosopher's stone in liquid form. Gluttony, Envy, Lust, Sloth, and Pride are all created by manufacturing the human bodies in which they reside, and fueling their bodies with souls from Father's philosopher's stone.
Greed (at least, the second one) and Wrath are both already human, and have souls put into them. In fact, Hohenheim and Greed-Ling do not differ in their abilities, essentially. They both have life extended by their philosopher's stones, and both have true, human bodies. Wrath is no different, except that he only has one soul within his stone, and so cannot regenerate, and also gets older with time.
Van Hohenheim, a human, unknowingly assisted Homunculus (the Dwarf in the Flask), in using the population of Xerxes to fill his own body with thousands of souls, while giving The Dwarf the false human body we see as Father. Father later uses his stone's powers to create the homunculi, the seven sins that we see. Greed-Ling and Wrath are just like Hohenheim, just having their own weaknesses.