In The Giver, what did Jonas give to Gabriel and why did he give it to him?
I think what you are talking about is how Jonas gives memories to Gabe. He does this for different reasons at different times.
At first, the reason Jonas gives Gabe memories is to help him sleep. Gabe is a restless baby and Jonas is afraid he will not be able to get placed with a family if he won't sleep well. If that happens, he'll be released. So Jonas gives him soothing memories.
Then later, when they are running away, Jonas gives Gabe memories of snow to make him colder so that the airplanes' infrared detectors won't be able to "see" him.
When Gabe comes to their house, Jonas realizes he is there to learn how to sleep through the night. If he doesn't, Jonas is told, he'll be released. Jonas sees something different about Gabe and wonders if Gabe can receive memories...and if Jonas can give them. So he asks his parents if Gabe can sleep in his room, and they agree.
He experiments with giving Gabe pleasant memories to help him sleep, and it works. He later gives him memories to keep him quiet as they are escaping and again to keep him warm as they are in the snowstorm.
Jonas gives "memories" to Gabe. He learns how to transmit memories. He first transmits memories to solve Gabe's sleeping problems at nights. The other time he gives something to Gabe is also "memories". He transmits memories to Gabe when they escape from the community, he transmits again some memories to prevent Gabe from the bad weather and hunger.