The following is a collection of some quotations which help to explain the Christian doctrine of the Trinity.
This first quote is from the 1689 Baptist Confession of Faith:
"Chapter 2: Of God and of the Holy Trinity
"1. The Lord our God is but one only living and true God; whose subsistence is in and of himself, infinite in being and perfection; whose essence cannot be comprehended by any but himself; a most pure spirit, invisible, without body, parts, or passions, who only hath immortality, dwelling in the light which no man can approach unto; who is immutable, immense, eternal, incomprehensible, almighty, every way infinite, most holy, most wise, most free, most absolute; working all things according to the counsel of his own immutable and most righteous will for his own glory; most loving, gracious, merciful, long-suffering, abundant in goodness and truth, forgiving iniquity, transgression, and sin; the rewarder of them that diligently seek him, and withal most just and terrible in his judgments, hating all sin, and who will by no means clear the guilty. ( 1 Corinthians 8:4, 6; Deuteronomy 6:4; Jeremiah 10:10; Isaiah 48:12; Exodus 3:14; John 4:24; 1 Timothy 1:17; Deuteronomy 4:15, 16; Malachi 3:6; 1 Kings 8:27; Jeremiah 23:23; Psalms 90:2; Genesis 17:1; Isaiah 6:3; Psalms 115:3; Isaiah 46:10; Proverbs 16:4; Romans 11:36; Exodus 34:6, 7; Hebrews 11:6; Nehemiah 9:32, 33; Psalms 5:5, 6; Exodus 34:7; Nahum 1:2, 3 )
"2. God, having all life, glory, goodness, blessedness, in and of himself, is alone in and unto himself all-sufficient, not standing in need of any creature which he hath made, nor deriving any glory from them, but only manifesting his own glory in, by, unto, and upon them; he is the alone fountain of all being, of whom, through whom, and to whom are all things, and he hath most sovereign dominion over all creatures, to do by them, for them, or upon them, whatsoever himself pleaseth; in his sight all things are open and manifest, his knowledge is infinite, infallible, and independent upon the creature, so as nothing is to him contingent or uncertain; he is most holy in all his counsels, in all his works, and in all his commands; to him is due from angels and men, whatsoever worship, service, or obedience, as creatures they owe unto the Creator, and whatever he is further pleased to require of them. ( John 5:26; Psalms 148:13; Psalms 119:68; Job 22:2, 3; Romans 11:34-36; Daniel 4:25, 34, 35; Hebrews 4:13; Ezekiel 11:5; Acts 15:18; Psalms 145:17; Revelation 5:12-14 )
"3. In this divine and infinite Being there are three subsistences, the Father, the Word or Son, and Holy Spirit, of one substance, power, and eternity, each having the whole divine essence, yet the essence undivided: the Father is of none, neither begotten nor proceeding; the Son is eternally begotten of the Father; the Holy Spirit proceeding from the Father and the Son; all infinite, without beginning, therefore but one God, who is not to be divided in nature and being, but distinguished by several peculiar relative properties and personal relations; which doctrine of the Trinity is the foundation of all our communion with God, and comfortable dependence on him. ( 1 John 5:7; Matthew 28:19; 2 Corinthians 13:14; Exodus 3:14; John 14:11; 1 Corinthians 8:6; John 1:14,18; John 15:26; Galatians 4:6 )" (As quoted at http://www.prbc.org/Confession.htm#Chapter 2)
The next quote is from Systematic Theology by Louis Berkhof, a Reformed systematic theologian:
"a. There is in the Divine Being but one indivisible essence (ousia, essentia). [...]
"b. In this one Divine Being there are three Persons or individual subsistences, Father, Son, and Holy Spirit. [...] To denote these distinctions in the Godhead, Greek writers generally employed the term hupostasis, while Latin authors used the term persona, and sometimes substantia. Because the former was apt to be misleading and the latter was ambiguous, the Schoolmen coined the word subsistentia. The variety of the terms used points to the fact that their inadequacy was always felt. It is generally admitted that the word 'person' is but an imperfect expression of the idea. In common parlance it denotes a separate rational and moral individual, possessed of self-consciousness, and conscious of his identity amid all changes. Experience teaches that where you have a person, you also have a distinct individual essence. Every person is a distinct and separate individual, in whom human nature is individualized. But in God there are no three individuals alongside of, and separate from, one another, but only personal self-distinctions within the Divine essence, which is not only generically, but also numerically, one. Consequently many preferred to speak of three hypostases in God, three different modes, not of manifestation, as Sabellius taught, but of existence or subsistence. Thus Calvin says: 'By person, then, I mean a subsistence in the Divine essence. -- a subsistence which, while related to the other two, is distinguished from them by incommunicable properties.'1 This is perfectly permissible and may ward off misunderstanding, but should not cause us to lose sight of the fact that the self-distinctions in the Divine Being imply an 'I' and 'Thou' and 'He,' in the Being of God, which assume personal relations to one another. [...]
"c. The whole undivided essence of God belongs equally to each of the three persons. This means that the divine essence is not divided among the three persons, but is wholly with all its perfection in each one of the persons, so that they have a numerical unity of essence. The divine nature is distinguished from the human nature in that it can subsist wholly and indivisibly in more than one person. While three persons among men have only a specific unity of nature or essence, that is, share in the same kind of nature or essence, the persons in the Godhead have a numerical unity of essence, that is, possess the identical essence. Human nature or essence may be regarded as a species, of which each man has an individual part, so that there is a specific (from species) unity; but the divine nature is indivisible and therefore identical in the persons of the Godhead. It is numerically one and the same, and therefore the unity of the essence in the persons is a numerical unity. From this it follows that the divine essence is not an independent existence alongside of the three persons. It has no existence outside of and apart from the three persons. If it did, there be no true unity, but a division that would lead into tetratheism. The personal distinction is one within the divine essence. This has, as it is usually termed, three modes of subsistence. [...]" (Italics in original. http://books.google.com/books?id=jFqJaODKwIoC&printsec=frontcover&source=gbs_v2_summary_r&cad=0)
The following quote is from Christian theologian Alister McGrath's Christian Theology: An Introduction, which is "one of the most widely used textbooks in Christian theology":
[...] "The three persons of the Trinity are distinct, yet not divided (distincti non divisi), different yet not separate or independent of each other (discreti non separati). [...]" (Italics in original. http://books.google.com/books?id=tHlY94UWi3UC&printsec=frontcover&source=gbs_v2_summary_r&cad=0)
And the following is from the official website of the Wisconsin Evangelical Lutheran Synod:
"The Christian Church confesses that there is one God, not three; that there are three distinct Persons in the Godhead, and that the Father is God, the Son is God, and the Holy Spirit is God. One God, three Persons.
"The Persons are distinct but not separate: God is One and undivided. There are not three 'parts' of God or a 'threefold' God. God is triune--three Persons in one God--not 'triplex' (made up of three gods). [...]" (http://www.wels.net/cgi-bin/site.pl?1518&cuTopic_topicID=399&cuItem_itemID=22530)