If you notice all the above responses the background is darkened by shadow or in other ways, while the subject is more brightly lit. There are various ways to do that.
In this first case, the subject was strongly backlit by the sun and the background behind it was in heavy shade. I used a very long telephoto lens (500mm) to make the background, about 15 or 20 feet away, blur down completely (it was an old weathered fence and wall).
This orchid was a different challenge. I shot it indoors with a 100mm macro lens by available window light ("North light", which is ideal), and didn't have much room behind it. So I hung a dark purplish piece of velvet behind the flower.
Here's another example where I hung a piece of black velvet behind the subject (to cover a gray stucco wall). In this case, I used just a little fill flash... hardly noticeable in the image.
For this shot of a garden spider, the natural background of distant trees in shade worked great. The subject and its web were backlit by the sun. I had to add stronger fill flash to brighten them up for the image.
All the above were done using a tripod.
This shot of a preying mantis wasn't done with a tripod. Instead I rested the camera and a 100mm macro lens on an upside down, 5 gallon paint bucket. Here I used a large, powerful flash with strong diffusion on it, held near the subject, as the primary light source. "Full flash", instead of fill flash. With my cameras, this is accomplished by using AE flash, but taking the cameras out of any auto exposure mode... using strictly manual exposure mode (no Auto ISO!) and being sure that the shutter speed, ISO and aperture make for a strongly underexposed background, while the flash insures the subject is well exposed.
I also just look for situations where subjects are nicely lit, against darker backgrounds. All these were simply shot hand held in various locations where things just worked out well...
All the above examples are macro or close-up shots... but much the same can be done with larger subjects, too.
Here's a portrait of a friend teaching a photography class, shot in his studio using only a single light source off to one side...
I simply under-exposed this image of a car's grill and hood... setting my exposure for the highlights:
A friend's cat simply posed well lit near a window, so the white wall much less well lit and some distance behind her went nearly black. I had to do a little retouching of the background in Photoshop... but not much.
Yes, it's possible to select a subject and "drop out" a background in Photoshop. It's just a lot of work and, in my opinion, often doesn't look "quite right"... seems unnatural or something. I'm a big fan of Photoshop, but in this case I prefer to "get it right" or close to it, in the camera when I take the shot.
A Circular Polarizing filter can help in some situations. A Neutral Density may be needed in others.