The simplest way I have to explain this is to look at an intrinsic quality of God (a quality that doesn't depend on anything else, or being compared to anyone) is Love.
First, good or powerful is not an intrinsic quality as that is a relative quality, in that God is good compared to Satan but perhaps not compared to Christ.
OK, Love requires three things: a lover, a receiver of the love and the relationship between them.
So, God the Father is the lover, Jesus is the recipient (the beloved) and the Holy Spirit is the manifestation of their love.
So, though they are three distinct beings, they are one God also.
This is a simple way to explain something that is beyond our ability to understand based on reason alone, as explained by Thomas Aquinas in Summa Theologica, and a good starting point is Book 1, Question 32, Article 1: http://www.newadvent.org/summa/1032.htmWhether the trinity of the divine persons can be known by natural reason?Squi.
To look at what Aquinas wrote regarding the Son and the Father you can look at Question 33: http://www.newadvent.org/summa/1033.htmThe person of the Father
I reference Thomas Aquinas often as he has a fantastic approach to using reason to look at these. This question in his book deals with your question: http://www.newadvent.org/summa/1093.htmThe person of the Father.
An image is not equal to the original, as, if you look at a mirror, you seen an image, but it is flat, and the original person is not, so you just see a small part of the person.
Unfortunately, anything I can write to answer this will go back to Aquinas, as his answer is probably the most complete starting point, as he looks at not only the image of God in man, but whether image and likeness are different and in between he looks at other issues on this subject.