Camera lens for great depth of field (Nikon)?

I have a Nikon D3100 and I'm looking for a lens that can make backgrounds extremely blurry.
If possible not too expensive. Preferably for closeups, doesn't really matter!

Added (1). Hehe whoops, sorry! Got them mixed up!

Depth of field is the sharp part of the photo… So if you need LESS deep of field you have to search a very "fast" lens ( f:2,8 o less ) that can give you a blurry b.ground… If you use it properly… Something like the 50/1,8G or 40/2,8macro

focal lenght is important too, but long lenses are ( too ) expensive

It seems that you do not understand the fundamentals of what we call depth of field

Great depth of field makes everything from near the camera to infinity, appear to be in focus. A lens like the 12-24 mm Nikkor on your D3100 would do that.

But you are asking about a lens that makes the background appear to be out of focus and that is called shallow depth of field. This is what produces the shallowest depth of field

* medium to long telephoto lenses
* lenses with wide apertures, like from f/1.4 to f/2.8
* placing the subject as close to the lens as it will focus.

And finally, closeups are in the realm of macro photography and the same applies, the closer the lens is to the subject, the wider open the lens aperture is, the shallower the depth of field, but when shooting with a macro lens, the depth of field can be as shallow as 1 mm, even less.

If on the other hand you want great depth of field when shooting close up with a macro lens, then you need to stop the lens down as far as it will go, f/22 or f/32. Small apertures like that cause a greater apparent depth of field.

As far as expense is concerned, you will have to make a choice. Either deal with depth of field with the lens you have now or start saving your pennies.

Here are some examples of lenses which can produce shallow depth of field, sometimes called selective depth of field when the photographer selects exactly what they want to appear in focus and what they do not.

AF-S Nikkor 12-24 mm f/4. This lens will create image that will look to be in focus from the closest object in the shot to the mountains, far a way if you shoot at f/22: $1,225

AF-S Nikkor 50 mm f/1.8. This lens is a favorite of portrait photographers because they can control the depth of field to include the subjects ears all the way up to the tip of the nose, all else will be out of focus: $220

AF-S Nikkor 105 mm f/2.8 macro. This lens is is a favorite for shooting close ups (from 1:1, 1:2, 1:3 and 1:4 image to subject ratio) as well as a general purpose medium telephoto lens: $985

For those who want to shoot sports under the lights or indoors, the AF-S Nikkor 300 mm f/2.8 is one you will see on almost every sideline: $5,900

There's the single lens solution popular for many advanced amateurs and those who shoot sports like motocross, the 18-200 mm f/3.5~f/5.6: $600

As you can see, before you can choose a lens for your fine D3100, you have to know what subjects you will be shooting and how dedicated you are to expanding your camera system.

With these lenses in mind, make some choices and start saving your pennies. In the mean time, you can learn a whole lot using the 18-55 mm lens on your camera. After all, once you are shooting your camera intuitively, when you do buy your next lens, you will be well on your way to producing brilliant shots, each time you press the shutter release.