Although most explanations of the Trinity fall short, here is a couple that I found helpful. Please forgive the lack of sources as this was gathered from a forum I used to attend and searching for the actual posts is a pain.
The Trinity from a union perspective:
God designed the union of husband and wife to teach us about the inner life of the Trinity. When we see the love of husband and wife overflowing into the fruitfulness of children, we learn a very important truth about God: God is not a sterile duality, but a fruitful Trinity. In the Trinity, the Two become One and so burst forth in a third Person. So, too, in the world, husband and wife are called to become one and give new life to another person. A family is an historical trinity reflecting the eternal Trinity.
Imagine a mountain. Upon this mountain is a moor. The weather patterns around the mountain change, and the moor is filled with water. From the moment the moor water forms into an aquifer, the aquifer produces a river. The river flows down the mountain, and nourishes the fields of wheat below.
Now, the aquifer is the source of the river. The moment the aquifer became an aquifer, it produced the river. That doesn't mean the aquifer isn't the source.
Similarly, the Father has always produced the Son and the Spirit - though that doesn't negate from the Father's status as the divine origin.
JamesThePersian (Eastern Orthodox):
God is One in His Divine Essence, His substance if you like, but He is made up of three Hypostases. Hypostasis is usually translated into English as Person, but that's a fairly poor translation, it's more like personal essence, that which makes an individual a unique person. There really isn't a good analogy that can be used because this is utterly other than all beings that we have experience of in real life. The best way I can describe it is this: a human has one essence (that which makes him human) and one hypostasis (that which makes him uniquely him). God has one Essence (that which makes Him Divine) but three Hypostases. He is, then, One God (one individual) in three Hypostases and is always, simultaneously, One according to His essence but Three in His Hypostases. I'm sure that my description fails at many levels, but we were asked to explain in our own words so I can't call on the words of the Fathers. It is impossible to really grasp the Trinity with our rational mind (hence us calling it a Mystery) so I'd be unsurprised if people don't follow my attempt to explain. You can pretty much guarantee that if someone tells you the Trinity is 'simple' they don't have a clear idea of what it means themselves.
From the Catechism of the Catholic Church:
253 The Trinity is One. We do not confess three Gods, but one God in three persons, the "consubstantial Trinity".83 The divine persons do not share the one divinity among themselves but each of them is God whole and entire: "The Father is that which the Son is, the Son that which the Father is, the Father and the Son that which the Holy Spirit is, i.e. by nature one God."84 In the words of the Fourth Lateran Council (1215), "Each of the persons is that supreme reality, viz., the divine substance, essence or nature."85
254 The divine persons are really distinct from one another. "God is one but not solitary."86 "Father", "Son", "Holy Spirit" are not simply names designating modalities of the divine being, for they are really distinct from one another: "He is not the Father who is the Son, nor is the Son he who is the Father, nor is the Holy Spirit he who is the Father or the Son."87 They are distinct from one another in their relations of origin: "It is the Father who generates, the Son who is begotten, and the Holy Spirit who proceeds."88 The divine Unity is Triune.
255 The divine persons are relative to one another. Because it does not divide the divine unity, the real distinction of the persons from one another resides solely in the relationships which relate them to one another: "In the relational names of the persons the Father is related to the Son, the Son to the Father, and the Holy Spirit to both. While they are called three persons in view of their relations, we believe in one nature or substance."89 Indeed "everything (in them) is one where there is no opposition of relationship."90 "Because of that unity the Father is wholly in the Son and wholly in the Holy Spirit; the Son is wholly in the Father and wholly in the Holy Spirit; the Holy Spirit is wholly in the Father and wholly in the Son."91