I'm looking to replace my tires.
Currently I have the standard 26x2.1 tires that came with the bike and am looking to replace them with a set of 26x2.25, could this be done?
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Tire sizing is essentially witchcraft (See http://sheldonbrown.com/tire-sizing.html) for mixing conventions and what not. Even if you have two tires of the same dimensions (say 26x2.25), you may not be able to mount both of them due to knob patterns (a "larger" labeled slick may fit where a "smaller" labeled knobby squeezes by due to the height and width of the knobs) or slight dishonesty in sizing preventing the frame from clearing things properly. Also, your rims need to be able to take the tire size as well. This all may also depend on the particular frame size you have as well. You should probably go to your LBS and try some 26x2.25's in the space or at least do some measurements on the clearance of the tire in the frame (it may not be the same in the front and back, so you may be able to run a larger tire in the front).
It seems to be a Halford's bike (which is like REI level bikes? not from teh UK), so there aren't product pages for it archived, so I don't have any specific sizing information I can help with. http://www.bikeradar.com/us/gear/category/bikes/mountain-bikes/product/review-carrera-vulcan-12-45673/ says they fit a Continental Mountain King 26x2.2 on there in the comments but doesn't mention how much more it will clear (which you'll have to actually go out and try similar tires as to the one you want).
From what I've been able to find, your Carrera Vulcan is probably equipped with a Suntour XCM V3. Since this is a 100mm (stated) travel fork, it's most likely intended for XC riding and will fit anything 2.3" and smaller. I've run 29x2.25" with a moderately aggresive knob pattern and didn't have any problems with clearance (on a Suntour XCT fork). However, if you're looking at something like an All-Mountain or Downhill type tire, I would go smaller as clearance will probably be tight.
So the short answer is: probably. You can also measure the most narrow distance between the legs of the fork lowers and see what that is and go from there.