Shutter speed very slow while using aperture mode (Nikon D3100)?

I'm new to professional cameras, and I'm trying to use aperture mode to try make the background of the picture a little blurry but when i use aperture mode, the shutter takes about 3-5 seconds to take the picture and the picture ends up blurry… I'm using Nikon D3100

First off, the D3100 is an entry level DSLR. Not all DSLRs are professional ones. Those start at about $1300, others may even say $2500.

The problem here is insufficient light. Any change to the aperture needs to be counteracted by changed to shutter speed or ISO. The camera does she shutter speed for you in aperture priority mode, and also the ISO if the ISO automatic is activated.
So you need to either activate the ISO auto to boost the ISO to get a proper exposure, or increase the ISO manually, or the camera will use way too long shutter speeds for a steady shot.

Probably you are using high f-numbers though, which means that you're using a wide depth of field. You get a blurred background by using low f-numbers, which give you a shallow depth of field. That will also increase the amount of light that reaches the sensor, and thus give you a faster shutter speed.

This article explains the basics. The controls on the camera are a bit different than yours (e.g.a physical switch on the camera for shutter speed, unlike your electronic controls), but the concepts are the same:
http://www.photographyjam.com/articles/29/camera-basics-shutter-speed-aperture-and-iso

Sounds like you don't have enough light. You could increase your ISO (how sensitive the sensor is to light), but increase it too much and the picture will be noisy!
If you are going for a shallow depth of field shot, you will already have your aperture pretty wide (lowest number), so you may need to add light, with flash or lamps.

Are you adjusting the aperture correctly? To get a shallow depth of field (background blurry) you need a wide aperture (that mean the f number must be small, eg f1.8 or 2.8). Some lenses can't go this wide - the kit lens for example has a maximum aperture of f3.5 at 18mm and f5.6 at 55mm so you can't get any wider than this.

The slow shutter speed you are getting is because there's not enough light - are you indoors? The picture is then blurry because the camera is moving during the exposure. If the aperture is the widest you can get and you still don't have enough light, then you have to add some - ie use flash. Or you buy a lens which has a wider aperture (ie works better in low light) like the 50mm f1.8.