There was once a time in AE where you had to precompose and/or nest layers to group them but no longer! A parent layer assigns its transformations (all except opacity) to its child layer. The child is still independent of the parent though, and can have its own animations applied.
Think of it like your right arm (or left, if you prefer). Your fingers can wiggle on their own, yet they are parented to your hand. If the hand moves, the fingers move. Likewise, the hand is parented to the arm, if it were to swing outward, the hand (and fingers) would follow. And lastly, the arm is parented to your body. Where the body goes, the arm goes, and hence the hand, and likewise the fingers. One parent can control many layers but a child can only have one parent.
If you don't see a Parent Column in the timeline, contextual click on the top of any column and choose Columns>Parent.
Use the Pickwhip to drag and select, or manually choose a parent by clicking on None.
If you are assigning a parent to many layers, select multiple layers and select a parent for one of them. The parenting will be applied to all selected layers.
Parenting can not be animated. If you want to stop parenting at any time, you need to split the layer, and assign None as a parent to the split.