I think many of the completely ignorant answers show exactly why it is so poorly misunderstood. The Trinity isn't irrational at all. However it is a revealed truth. There is simply anything like it in any other system of belief in the world. If you believe that pagan trinities of Gods are the same thing than that is simply a misunderstanding on your part.The Trinity teaches us everything about Judeo-Christinaity as the Trinity is a family unto Itself and Judeo-Christianity has to be understood through the successive covenants in which God, i.e. the Trinity, has attempted to bring us into his own family life, as intimates. From my perspective now as a convert to Catholicism, i.e. Christianity, the whole world and every family in the natural world, of both men and animals, is a reflection of this reality. The Trinity also leads to some very beautiful implications such as truth and love being one and the same thing--not only are they the same but they provide for a theory of objective truth/love which is grounded in God's ontological encounter with himself.
One thing to understand, and unfortunately psychology speaks a different language so this reality of our psychological being isn't reflected in secular psychological theory, is that there is a difference between persons and essences. For us our essence and person are pretty much the same thing so it is not immediately apparent that this could not be the case. However even we as humans, from a Christianity perspective still do share something of common essence with one another. This is what makes human communion, and therefore community, possible in the first place. It is also how we all came lose Sanctifying Grace through activity of single person and how we were restored to it through the actions of another single person.
Initially upon my investigation in Catholicism I thought the Trinity was sheer irrationality, and I would never be able to accept it. Once however I came to actually understand, in the partial way we are limited to understanding it, I saw that it must always be the logical consequence of a God in whose image we are made. I reiterate the simplest formula below:
God the source and ground of all Being exists, non contingently, in absolute simplicity as pure act. The self aware of God, is itself a person. This is not very different than our narratives of ourselves that constituted the sum total of our ego's in some uses of the term. Essentially our egotistic conception of ourselves is a story that defines us. In God's case this story is absolutely accurate, total and so like the himself infinity. God has perfect knowledge of His own infinite self and this is the begotten son, aka the Word of God. As the Son gazes back at the Father encountering his own ontological reality Truth and Love are both born and are themselves personified this is the Holy Ghost, and since all three persons of the one Essence are the same the Father and the Son are also both the incarnation of Love and Truth, itself. The three Persons of the Trinity life in total self sacrifice (love) to one another. Jesus who incarnated into the flesh and is also know and the Eternal Covenant made flesh taught us the depths of what such self giving means as he gave himself for us as St.Paul says while we were still in our sin.
The French post-structuralist Jacques Derrida wrote much in life about the impossibility of the pure gift. For humans it is impossible--at least without the grace received by saints--for us to give without some notion of economy, e.g. some notion of tit for tat, but Jesus gave himself in full acceptance of our ingratitude. In fact this is what marks the Trinity who has no need of anything outside of Itself, yet out of love and desire to share Itself created all creation. Here is where of the notion of the Covenant comes into play. A covenant is as different from a contract as marriage is from prostitution. A contract is an agreement between person for the exchange of objects or services. A convent invokes God, assuming he is real (this is a big deal because if God really exists than Covenants have real authority), and is an exchange of persons. Covenants create family bonds. God who is family, in and of Himself, wants nothing more than to make us family too. The Trinity is everywhere the Old and New Covenants (translated into Latin as Testament) of Sacred Scripture. The properly understood the very fact we divide salvation history in Old and New "Testaments", itself is logical implication of the Trinity.
From the above I hope it clear that you can't have God without consequently having the image of God, His Son, and where there is the Father and the Son there must then necessarily be the Holy Spirit, although I believe the term ghost captures a little more of the manifestation of this Third Person. Quite frankly a God,, in whose image we are made, who is not Triune is simply not logically feasible.
Why don't people better understand this? I don't know, but it is not the only thing people fail to realize about Christianity. Since my conversion I've started to call it the greatest open secret on the planet. The Apostle's who comprised the Church from the beginning took pains to teach in creeds our faith in creeds. Yet very little of what the Church has therefor consistently taught since the beginning is realized even among the majority of Catholics. One can blame the signal to noise ratio. It is as if some force in the world wants to so muddy the water that even people who sincerely search for it have difficulty seeing the bottom.