By all means use a layer mask . . . if you don't, once you erase something then the pixels are gone for good. By using a layer mask you can use the paintbrush tool to paint the pixels back in.
Step 1 in Zizi-K's post is very important. If you don't do that, you'll just be turning the background to white pixels . . . you won't be removing the background, and you won't be able to put just kitty on a different picture, you'll have kitty sitting in a white box (in this case Bethalize's suggestion will work.)
Once you have your photo on Layer 1, look at the bottom of the "Layers" tool box (Top Menu > Window > Layers) there is a small icon that looks like a square with a circle inside it. Click on that and you'll see another square show up next to Layer 1 square. This is your mask.
Make sure you have your mask selected when you delete something (click back and forth between the first square and the second, you'll be able to tell if the mask is selected or not.)
I agree with Zizi-K's' second point -- The magic wand tool is the easiest for beginners to use.
To expound on "play with the settings" . . . when you click on the magic wand tool, on the top menu bar you should get options.
Tolerance = chooses pixels by color/shade. The lower the number the smaller amount of pixels get selected. For instance, in the top part of your photo where there is a huge contrast between the white wall and the black kitty, you could probably get away with a tolerance of 32+. The gray shadow behind kitty's tail, you'll want to lower your tolerance to maybe 12. Yes, play with the settings to see what works for you.
You do want "Anti-alias" clicked.
Looking at the background you would also want "Contiguous" clicked, otherwise you might select the highlights in kitty's eyes when you click on the wall background since they are close to the same color.
The magic wand will not help you at all around the stool that kitty is sitting on. Even with a tolerance of 1, kitty's entire body will be selected because the stool and kitty's body are the same color.
This is where you play with your eraser and paint tools. Try different brush sizes along with different hardness settings (it's the same as the magic wand tool, once you click on the tool you'll get options on the top menu bar. Click on the brush option and you'll get a drop-down menu for brush size and hardness.) Yes, you can also use the polygon selection tool or the pen tool (my preferred, but a little more advanced.)
Delete a little bit at a time. Don't try to remove the entire background at once . . . your background is quite varied and a bit tricky at certain spots. (eta: "History" window is your friend . . . in photoshop you only get 1 "undo")
Again, make sure that you are on your layer mask. If you delete/erase something that you didn't want to delete/erase, switch to your paint tool and paint the pixels back in.
Additional tip: Just to make sure that you are, in fact, getting a transparent background go to "Preferences > Transparency & Gamut > set your grid size to small, medium, whatever . . . if it's set at "none" it looks like a white background.
Hope this helps . . . and have fun!