One click gets you 90% of the way, and the last 10% is a layer mask. Do not delete the background — mask it, so you can fix the edge afterwards.
Steps
Unlock the layer if it says Background: click the padlock in the Layers panel.
With the layer selected, use Select → Subject. Photoshop finds the main thing in the photo.
Click the Add layer mask button at the bottom of the Layers panel. The background disappears without anything being deleted.
Fix the edge: double-click the mask and use Select and Mask. The refine-edge brush along hair or fur is the part that makes it look real.
To correct by hand, click the mask thumbnail and paint on it — black hides, white brings back. That is the whole model.
Worth knowing
The quick action Remove Background in the Properties panel does all of the above in one click, but gives you the mask without the selection, so refining is the same job either way. Hair on a busy background is where automatic selection fails; a plain backdrop when you shoot saves an hour later.
Faster than searching: box the part of your screen you
cannot work out and ask about it directly — in Photoshop or anywhere
else. Answers arrive where you drew the box.