The Persons of the Trinity are not three individuals but three persons; we, because of our personal experiences (every person I've met today is also an individual), tend to consider them synonymous, and may think of there being three separate minds and wills in the Trinity. But the differences between the Persons of the Trinity are only in their relation toward one another; They share one mind and will, yet each Person is self-aware (the Father knows He is the Father, He knows the other two Persons, and He knows of the other two Persons as distinct yet one with Him; and so on).
The Godhead has one rational nature; each Person of the Trinity shares that rational nature in a distinct way.
The following quotations come from Karl Rahner's The Trinity, and anything in bold is my emphasis; italics are present in the original. Note that he ultimately teaches a heretical view of the Trinity due to his claim that creation was a necessary action on God's part and the generation of His Son was in preparation for creation. I'm not using any of that here.
Rahner agrees with Aquinas's definition of person: That which subsists as distinct in a rational nature (Summa Theologica, Q 30, A 4). Whereas we normally think that a person has a mind, by this definition it's the opposite: minds contain persons.
He begins with a high-level discussion of the relation between each Person of the Trinity, summarized here (also see this question):
It follows that we must say that the Father, Son, and Spirit are identical with the one godhead and are "relatively" distinct from one another. These three as distinct are constituted only by there relatedness to one another, so that the axiom which asserts the identity of the essence and the distinction of the three may also be formulated (as Anselm was the first to do and as done by the Council of Florence) as follows: in God everything is one except where there is relative opposition (DS 1330).
So the Father is fully God, the Son is fully God, and the Spirit is fully God; we do not have an additive relationship, where Father + Son + Spirit = God; neither do we have a single Person revealing Himself in three different forms, but we have three Persons "relatively" distinct. Each Person sort of comprises the "God essence" in a distinct way.
Rahner then begins to discuss the relations in more detail, preparing to take a closer look at each Person within the Trinity:
Let us first make it clear that in the official doctrine of the [Catholic] Church this term [person] says nothing which has not yet been said with the word "hypostasis." This is already evident from the fact that both words are used as synonyms. Furthermore, it is evident that the element of consciousness, which nowadays and from long ago almost spontaneously connected with the concept,28 does not belong to it in our context, insofar as it expresses the formal moment of this concept as distinguished from the essence of God. Otherwise the "three" would also have to be said of this "element of consciousness." But there exists in God only one power, one will, only one self-presence, a unique activity, a unique beatitude, and so forth.29 Hence self-awareness is not a moment which distinguishes the divine "persons" one from the other, even though each divine "person," as concrete, possesses a self-consciousness. Whatever would mean three "subjectivities" must be carefully kept away from the concept of person in the present context.20
28: This is also often tacitly, but wrongly, presupposed in some "demonstrations" by which biblical theology tries to establish the "personality" of the (immanent) Son and Spirit, when they wish to demonstrate a "personality" of the Son and Spirit which is distinct from that of the Father.
29: Cp. DS 3, 71, 73, 144f, 172, 177, 415, 421, 441, 451, 490, 501, 542, 545f, 572f, 680, 851, 3350.
We must, of course, say that Father, Son, and Spirit possess self-consciousness and that each one is aware of the other two "persons." But precisely this self-consciousness (as subjective, not as understood in its objectivity) comes from the divine essence, is common as one to the divine persons, is therefore a moment of the concrete person, so that he may be defined as "a distinct subject in a rational nature." But it is not a constitutive moment of the "person" as such, as distinct from the "essence" (nature), although it must be mentioned if we wish to explain the difference between a "hypostatis" with self-awareness and a subhuman thing-like "hypostatis."
30: Hence within the Trinity there is no reciprocal "Thou." The Son is the Father's self-utterance which should not in its turn be conceived as "uttering," and the Spirit is the "gift" which does not give in its turn. Jn. 17, 21; Gal. 4, 6; Rom. 8, 15 presuppose a creaturely starting point for the "Thou" addressed to the Father.
So each Person shares the Divine will, power, activity, and self-presence, all things that come from a mind. Yet each Person recognizes Himself as distinct from each other Person within the Trinity. Each Person's self-awareness and consciousness is not inherent to that Person (by nature of that Person being that Person) but comes from the shared essence.
Now this is not necessarily true; though this does explain the one-and-threeness without danger of imagining God as three Gods. We can only know of God what He chooses to reveal to us, and He didn't seem to think it necessary for us to fit Him into our little heads.
Nonetheless, the main difficulty regarding the concept of person in the doctrine of the Trinity is rather different, and we have already mentioned it several times before now: when today we speak of person in the plural, we think almost necessarily, because of the modern meaning of the word, of several spiritual centers of activity, of several subjectivities and liberties. But there are not three of these in God -- not only because in God there is only one essence, hence one absolute self-presence, but also because there is only one self-utterance of the Father, the Logos... There is in God a knowledge of these three persons.. a knowledge about the Trinity both as consciousness and as "object" of knowledge (as known). But there are not three consciousnesses; rather the one consciousness subsists in a threefold way. There is only one real consciousness in God, which is shared by the Father, Son, and Spirit, by each in his own proper way.