The current Wikipedia entry on Swedenborg contains an error
First, it should be noted that the statement on the current "Emanuel Swedenborg" Wikipedia page that "according to The Heavenly Doctrine, Jesus, the Son of God, came into the world . . . ." is not accurate. In Swedenborg's theology, the Son of God did not come into the world, nor does "Jesus" have the same meaning as "the Son of God."
The idea that the Son of God came into the world is based on the Nicene Trinity of Persons, which Swedenborg rejected. Rather, according to Swedenborg the Son of God came into being in time when God came into the world via the Incarnation. Here is how this happened, and what "the Son of God" means, according to Swedenborg:
The Lord as the Divine-Human One Is Called "The Son of God" and as the Word Is Called "The Son of Humanity"
The church knows only that the Son of God is the second person of the Godhead, distinct from the person of the Father, which results in a belief in a Son of God born from eternity. Since this is everywhere accepted and is about God, there is neither ability nor permission to think about this matter at all intelligently, not even about what it means to be "born from eternity." This is because people who think about it intelligently inevitably find themselves saying, "This is completely beyond me. Still, I say it because everybody else says it, and I believe it because everybody else believes it." They should realize, though, that there is no Son from eternity; rather, the Lord is from eternity. Only when they realize what “the Lord” means and what "the Son" means can they think intelligently about a triune God.
 As for the fact that the Lord’s human side—conceived by Jehovah the Father and born of the Virgin Mary—was the Son of God, this is obvious from the following in Luke:
The angel Gabriel was sent by God to a city in Galilee named Nazareth, to a virgin betrothed to a man whose name was Joseph, from the house of David. The virgin’s name was Mary. Having come in, the angel said to her, "Greetings, you who have attained grace. The Lord is with you; you are blessed among women." When she saw him, she was troubled by what he said and considered what kind of salutation this was. The angel said to her, "Do not be afraid, Mary: you have found favor with God. Behold, you will conceive and bear a Son, and you will call his name Jesus. He will be great and will be called the Son of the Highest." But Mary said to the angel, "How will this take place, since I have not had intercourse?" The angel replied and said to her, "The Holy Spirit will descend upon you, and the power of the Highest will cover you; therefore the Holy One that is born from you will be called the Son of God." (Luke 1:26–35)
This passage says "you will conceive and bear a Son. He will be great and will be called the Son of the Highest," and again, "The Holy One that is born from you will be called the Son of God." We can see from this that it is the human nature conceived by God and born of the Virgin Mary that is called "the Son of God." (Teachings for the New Jerusalem on the Lord, #19)
So you can see that in Swedenborg's theology, the Son of God did not come into the world, as in Nicene trinitarian theology. Rather, the Son of God came into being when Mary conceived in her womb from the Holy Spirit—which, based on the resulting child having no human father, and being called "the Son of God," is clearly the activity of God impregnating Mary. The Son of God, then, is the human nature whose Father was God and whose mother was Mary.
This, however, changed over Jesus' lifetime because of his process of "glorification," in which he replaced the finite human nature that he received from Mary with an infinite divine human nature that is fully divine, and fully God. In Swedenborg's theology, after the Resurrection and Ascension, "the Son of God" refers to the Divine Humanity of the Lord.
Further, in Swedenborg's theology, Jesus never separated from his Father as humans separate from their fathers. Rather, the Father was Jesus' own inner divine self, or soul. This is why he said, for example:
Have I been with you all this time, Philip, and you still do not know me? Whoever has seen me has seen the Father. How can you say, 'Show us the Father'? Do you not believe that I am in the Father and the Father is in me? The words that I say to you I do not speak on my own, but the Father who dwells in me does the works. Believe me that I am in the Father and the Father is in me . . . . (John 14:9–11)
For more on Jesus' process of glorification, and for the Swedenborgian explanation of why Jesus sometimes prayed to the Father as if to a separate being, please see this question, and my answer to it:
How does the Swedenborgian Church explain passages where Jesus talks/prays to the Father?
According to Swedenborg, the risen and glorified Jesus Christ, whom Swedenborg refers to simply as "the Lord," is the name given to the entire Godhead, including Father, Son, and Holy Spirit. All three of these are in the Lord and are the Lord. Hence my statement above that in Swedenborg's theology, "Jesus" does not have the same meaning as "the Son of God." The Son of God is one part of Jesus Christ, who is our Lord and our God (compare John 20:28).
Incidentally, in the above translation "the Son of Humanity" is a more contemporary and more literal translation of the Greek (and Latin) traditionally translated "the Son of Man."
In Swedenborg's Trinity, Father, Son, and Holy Spirit are "essentials" of God
According to Swedenborg, Father, Son, and Holy Spirit are not Persons of God, but "essential components" (Latin: essentialia) of God.
The Latin word essentialia is difficult to translate into common English. Traditionally it is simply translated "essentials." It can be thought of as "the basic elements of God without which God is not God."
Here is a lengthy quotation in which Swedenborg explains his concept of the Trinity in one Person of God:
(1) There is a divine Trinity, which is the Father, the Son, and the Holy Spirit. It is very obvious in the Word that there is a divine Trinity, which is the Father, the Son, and the Holy Spirit. Take the following passages for example:
The angel Gabriel said to Mary, "The Holy Spirit will descend upon you and the power of the Highest will cover you; therefore the Holy One that is born from you will be called the Son of God. " (Luke 1:35)
Here three are named: the Highest (who is God the Father), the Holy Spirit, and the Son of God.
When Jesus was baptized, behold the heavens opened and John saw the Holy Spirit coming down like a dove upon him; and a voice from heaven said, "This is my beloved Son, in whom I am well pleased." (Matthew 3:16-17; Mark 1:10–11; John 1:32)
The Trinity is even more obvious from the Lord's words to his disciples:
Go out and make disciples of all nations, baptizing them in the name of the Father, the Son, and the Holy Spirit. (Matthew 28:19)
The Trinity is also obvious from these words in John:
There are three who testify in heaven: the Father, the Word, and the Holy Spirit. (1 John 5:7)
Further evidence besides these passages is that the Lord prayed to his Father and spoke about him and with him; and he said that he was going to send and had sent the Holy Spirit. Furthermore, in their letters the apostles frequently mention the Father, the Son, and the Holy Spirit. These sources clearly show that there is a divine Trinity, which is the Father, the Son, and the Holy Spirit. . . .
(2) These three, the Father, the Son, and the Holy Spirit, are three essential components of one God. They are one the way our soul, our body, and the things we do are one. In any given thing there are general essential components and there are also specific essential components. The general and specific components combine to make one essence.
In our case, our general essential components are our soul, our body, and the things we do. These three components combine to make one essence, as you can see from the fact that one component comes from and exists for the other in an unbroken chain. We begin from our soul. The soul is the essence of the semen that originates us. Our soul not only initiates but also sequentially produces the features of our body. Then there are the things we do, which come from both our soul and our body. Because one of these components produces another, and therefore the subsequent components are grafted onto and connected to those that came before them, it follows that these three components share one essence. This is why they are called the three essential components.
The same three essential components—soul, body, and action—existed and still exist in the Lord God the Savior, as everyone acknowledges. The concept that the Lord's soul came from Jehovah the Father is something only the Antichrist could deny, since the Word of both testaments calls him the Son of Jehovah, the Son of God the Highest, and the only begotten One. The Lord's primary essential component, then, is the Father's divinity, like the soul in us. It follows that the Son whom Mary bore is the body of that divine soul; for what develops in the mother's womb is the body that was conceived by and derived from the soul. This, then, is the second essential component. Actions make a third essential component because they come from both the soul and the body; for things produced have the same essence as the things that produce them.
The three essential components that are Father, Son, and Holy Spirit are one in the Lord as our soul, our body, and our actions [are one in us]. This is clear and obvious from the Lord's statement that the Father and he are one, and that the Father is in him and he is in the Father. The Lord is also one with the Holy Spirit because the Holy Spirit is divinity radiating from the Lord on behalf of the Father, as I have fully shown from the Word in 153 and 154 above. To demonstrate this point again would therefore be an extra serving—it would be burdening the table with food after people are already full. (True Christianity #164, 166–167, links added)
In short, in Swedenborg's Trinity, Father, Son, and Holy Spirit are not Persons of God. Rather, they are what might loosely be called fundamental parts of God in the same way that our soul, body, and actions are fundamental parts of us.
After all, according to Genesis 1:26–27, God created humans in the image and likeness of God. If there is a Trinity in God, and we are created in the image and likeness of God, then there must be a trinity in us as well. And we know that we as human individuals are not three persons, but one person in which there are three parts: our soul, our body, and our actions. Our soul is our inner self, our body is our outer self, and our actions are the activity of our body from our soul, without which we cannot be said to be a human being.
In the very same way, the Father is God's soul, the Son is God's body or human presence, and the Holy Spirit is all of God's words and actions flowing out into the universe, and into human hearts and minds.