Why do my pedals move when I move my wheel in both directions? How do I fix it? Whichever direction I rotate my pedals, the wheel rotates. And when I move the wheel, same thing. How do I fix it?

Here is a clip: 
                
            

This is the model: 
http://www.halfords.com/webapp/wcs/stores/servlet/product_storeId_10001_catalogId_10151_productId_786803_langId_-1_categoryId_165499
This bike definitely has a derailleur and is not intended to operate as a fixie (i.e. something is definitely broken.) 

A normally functioning multi-speed bicycle has a mechanism that allows it to pedal forward and coast (freewheel) when standing on the pedals. Many older or cheaper bicycles have a "freewheel" which accomplishes this. Other bicycles have a freewheel integrated into the hub body, a "freehub." They're essentially the same mechanism - a ratchet with a few pawls.

In all likelihood, the pawls inside of the freehub or inside of the freewheel have seized or have come apart. Procure a can of penetrating oil and spray the seam where the cogs and the hub meet (i.e. where it is supposed to rotate. I've included a photo with arrows indicating where to spray. Be careful not to get oil on your rims or brake pads.) 

If this doesn't loosen it up, you'll need to either replace the freewheel or overhaul/replace your rear hub.