This is a really tricky question, isn’t it! Many Christians have puzzled over this for many years, so it is pretty normal to struggle to understand how the relationship between God and Jesus works. I’m going to try to handle your questions in reverse order if that’s ok.
So the Bible tells us that Jesus isn’t ‘God’s human son’, rather, that Jesus, who is co-eternal with God (that means that like God, he existed forever) became human. Jesus was like God – eternal and perfect – but he became human so that he could enter into our world and die for humans. The Bible treats this as something to marvel at: … Jesus Christ ‘though he was in the form of God, did not count equality with God a thing to be grasped, but made himself nothing, taking the form of a servant, being born in the likeness of men. And being found in human form, he humbled himself by becoming obedient to the point of death, even death on a cross.’ (Philippians 2:6-8) The actual ‘mechanics’ of how God had a human son are not explained to us, but we are left in no doubt that Mary’s conception and pregnancy are miraculous – you could read chapter 1 of the gospel of Luke to see the angel tell Mary that she will fall pregnant. The gospel of John, which is written in very symbolic language at the beginning, speaks about ‘the word becoming flesh’. ‘The word’ is Jesus – who was with God in the beginning – and he became flesh – this is what ‘incarnate’ means – to be made flesh. This makes it clear that Jesus is the eternal God becoming human. So in answer to your second question, it isn’t so much that God had a human son, but that God’s son became human for our sake.
So, to look at the first part of your question, what does it mean for God to have a son? Well, actually, when we say ‘God’ we could almost mean three people – God the Father, God the Son, or God the Holy Spirit. These three are called ‘the trinity’ and each one is called a ‘person’ of the trinity. It is quite hard to understand how these three work together. Some people find it easier to understand with an analogy – so, for example, God is like water – ice, water and steam are all water but they are all different. The analogy fails at various points but that might be a way to start to get your head around it. So a few things about the trinity that we should keep in mind:
* There is only one God – so even as we talk about the three persons of the trinity we are talking about one God. All three persons of the trinity are God. If you want to look at some verses, you could look at Deuteronomy 6:4, Galatians 1:1, John 1:1-18, and Matthew 28:19.
So the three persons of the trinity are the same God, but they are each distinct. They have different roles, but each action any member of the trinity might do is God’s action, regardless of who did it.
* We see this distinction / unity when Jesus identifies himself with the Father, saying that he and the Father ‘are one’ (John 10:38, 17:11,21), and that he is in the Father and vice versa (John 14:11). Jesus does not say that he IS the Father, or that he and the Father are the SAME, but that they are ONE. So they are distinct, yet unified.
* Because Jesus and the Father are both one and distinct, we can say that God ‘sent his son into the world’ – (John 3:16), and also that Jesus came into the world (1 Tim 1:15) – we are speaking about essentially the same action on the part of God, but on the one hand, God the Father sends, on the other, God the Son comes.
*We also see the distinction / unity at work when Jesus speaks about sending the Holy Spirit in John 16 – he describes this as both he and the Father coming. So the three persons of the trinity are distinct, but they are all God – they all share the same motivations and objectives. They are all co-eternal, they are all perfect.
God is Father, Son, and Holy Spirit – one God, existing in perfect relationship within himself. This is how God can be both Father and Son – because he just is! It’s a bit unsatisfying – to not wrap our heads around it more – but if we want to have a right idea of who God is and what he is like, it’s necessary to hold these strands together.
I hope this goes some way to answering your question. A really helpful thing to do would be to read John 14-16, where Jesus explains some of this himself. In fact, why not read John from the beginning to get the picture of the whole gospel and how it builds together our picture of the relationship of the Son to the Father, especially as the Son came to earth as a man to reveal the Father to us. These are hard things to understand but well worth the mental effort!
I wish you well in your efforts to understand these great things!