There are a few ways to remove the white background from an image depending on what your trying to achieve or the colours found on the image itself.
Here are a few ways of achieving your requested outcome.
- Magic Wand Tool
Select the layer of the background you would like to remove and select the Magic Wand Tool (Shortkey: w) then once selected you want to simply just left click on the white background and it should create a selection over it. (Note: if you want to change the amount of white that is selected there is an option in the toolbar to change the 'Tolerance') After you have selected it i personally would make a layermask so its not irreversible later to do that simply click the 'Add a Mask' button found in the layers Panel. Alternatively you can just hit delete and it removes the white by erasing it.
- Blending Options
This option is useful if there is no other white in the image or lighter colours, right click on your layer with the white and hit blending options once in the blending options you will find the panel at the bottom of the pop up that says 'Blend if Grey' and all you have to do is drag the sliders from the white side closer to the black as shown in the image below.
To have more control you can split the arrow pointing upwards in two like i have to do this just press Alt whilst selecting it this allows you to create a better blend.
- Pen Tool
This one is pretty simple and probably the best/cleanest option as you have full control of the situation. Select the pen tool and trace around the area you would like to remove and once done right click choose 'Make Selection' and again either make a mask Layer or simply just hit delete all this has to be done in the layer of which you want the background removed. For help on Pen tooling here is a little guide (guide:https://design.tutsplus.com/tutorials/photoshops-pen-tool-the-comprehensive-guide--psd-718)
- Layer Blend Modes
You can also sometimes get away with using blend modes i personally don't suggest this as anything below your image can occasionally pierce through and cause issues later down the line but if your comfortable and not doing something to complex this can be an option. To do this simply select the layer your working on and in the layers panel choose 'Darken' or 'Multiply' (see image) this will only show the dark area's of the image and hide the lighter ones alternatively you can do the opposite with 'Lighten' in case you ever want to just show the light areas in a layer and hide the dark