Why does my layer keep moving behind other layers?

Please drag your screenshots to the reply field instead of the Drag and Drop button so we can see them without downloading them.

If you are new, spend the first half hour with the Learn workspace so you can figure out the UI. Spending a couple of hours with the User Guide would also be a good idea. I don't know anyone that has been able to jump into AE and figure out how to create an animation or a composite shot by just poking around.

Your screenshots show a nested comp (Pre-comp). The Properties panel tells me that there is a huge difference in the Position values of Layer 1 in the two screenshots, which explains why the layer looks different in the Composition panel. If there are 3D layers in the nested comp, turning on Collapse Transformations will also make a big difference in how the comp looks because the Comp Camera in the main comp will render the 3D layers in the nested comp. 

Please explain what you are trying to do. I see Position and Point of Interest keyframes on the camera layer (Layer 2). The Y Point of Interest value changes in Screenshots 1 and 2. POI differences and layer position differences both contribute to the different positions of Layer 1 in the comp. I see nothing I do not expect to see.

If you are following a tutorial, please post a link so we can understand your workflow.

Your screengrabs don't give away much, because they only show one layer, and you haven't revealed the Position parameter in the timeline so we can't see where the layer is positioned in space.

After Effects 2D layers appear in the order they're stacked in the timeline.  Layers at the top of the stack appear on top of layers at the bottom.  Just as they do in Photoshop, Illustrator, Premiere Pro etc.

After Effects 3D layers have X Y and Z position values, allowing you to position them in 3D space.  A layer's position in space determines whether its in front of another layer, rather than its level in the layer stack.  So, assuming your 3D camera is in a conventional position in front of your layers,  a layer with a Z value of 200 will appear behind a layer with a Z value of 100.  A layer with a Z value of 10,000 will be way off in the distance, while a layer with a Z space value of -200 may be so far forward that you (or the 3D camera) can't even see it. 

As soon as you switch your layers to 3D, you have to think about their position in 3D space, rather than their hierarchy in the timeline layer stack.

Correction, picture 1 is the final result if I move things. Picture 2, is when it's just been copied and still appears as the top layer.