Not three distinct units (as there would be in a clover, say) but three distinct relations of origin.
What distinguishes the Father from the son is His Fatherhood and the Son’s Sonship. What distinguishes the Holy Spirit from the Father and the Son is the Spiration (on the part of the Father and the Son) and the Procession (on the part of the Holy Spirit).
Because we are talking about supernatural realities, it is impossible to understand them in any way, save by analogy with natural things, but we have to keep in mind that those analogies are very improper: by them we learn much more about what God
isn’t than about what He
is.
Anyway, we can look at the Trinity like this: the Father’s relation of origin consists in communicating the Divine Essence to the Son. We have to keep in mind, however, that the Divine Essence is utterly one, so it must be communicated wholly and entirely.
That means, among other things, that it is not as if the Father and the Son are members of the same “species”—as is the case between a human father and a human son. Rather, they are perfectly identical in Essence (or Substance): that is why we say in the Creed that the Son is “consubstantial with the Father.”
This communication of the Divine Essence could be compared to the way that the human mind makes concepts and judgments: a sort of likeness to the the thing known is generated in the mind. The difference is that, the only proper object to God’s knowledge is Himself (He knows his creation by means of His Divine Essence); moreover, when God knows HImself, the “concept” he “forms” (not that He actually “produces” a concept as we do) is not just a likeness, but the very Essence itself. So complete is that self-knoweldge, that it is a Person, like the Father.
The
only difference between the two Persons is that the Father begets, and the Son is begotten. Everything else, so to speak, is perfectly identical, for it is the very Divine Essence. (It would be better to say, the very Divine Essence is such that the Father begets the Son eternally.)
Similarly, the Father and Son, who know each other so intimately (being the very same Divine Essence) also love each other with a love that we cannot imagine. The way the Eastern Fathers describe it, they “sigh” for each other, and this “breathing” (spiration) results in a second communication of the Divine Essence. In other words, their mutual love is so compelling that it is another Person, the Holy Spirit.
In communicating the Divine Essence to the Son, the Father “loses” nothing; and when the Father communicates the Divine Essence through the Son to the Holy Spirit, neither Father nor Son “lose” anything. The Divine Essence remains utterly simple.
I don’t know if that helped or made it more difficult to understand

. If you are curious, you can take a look at St. Thomas Aquinas’ fabulous treatise on the Trinity in
Summa theologiae I, qq. 27-43.