Adoptionism is like when someone adopts a pet or a child. Let me explain this a little more, imagine a stray kitten that doesn't have a home or family to take care of it. If a person decides to take that kitten home and love and care for it, that person is adopting the kitten. In the same way, adoptionism is a belief that God adopted Jesus as his son, rather than Jesus being born as the son of God.
So, when we say "Jesus is the son of God" we usually understand it to mean that he was born as God's son. But, in adoptionism, it is believed that Jesus was a regular person who lived a good life and then God adopted him and made him his son. It's like God saying "I really love this guy, he's doing great things, so now he's my son."
This belief was popular among some early Christians but it is not widely accepted by most modern Christian denominations.