The mm is millimeters (length of lens from front element to focal plane (where the sensor is if I remember correctly. On a 35mm camera (the 35mm there is the film size) a 50mm lens is roughly equivilant to the same angle of view your eyes have naturally. The Shorter the lens (ie 24mm is shorter than 50mm) the wider the angle of view. The Longer the lens, the smaller the angle of view (more magnification).
On most DSLR's, because the sensors are smaller than the film they are replacing, there is a "Crop Factor" (also refered to as a 1.5x magnification, but that's not really true... potAto/potato kind of thing)
Nikon uses a "DX" size sensor which equates to a 1.5x crop factor magnification which is why they say the 70-300mm = 105 to 450mm on a Nikon Digital. On your Nikon Digital a "natural" view can be had with a 35mm lens.
This lens should be a great lens. Nikon claims the VR is good for 4 stops of vibration reduction, it has the silent wave motor (FAST autofocus, and you can refocus any time you want manually). The one thing you might be disappointed in is that it is a little long for the puppy photography. A lens of this length would be good for head and shoulders portraits from about 8 to 10 feet.
If you can find it, spend the extra $200 or so to buy the 18-200mm VR lens from Nikon. It is AWESOME, but is hard to come by, and it has a much better usable range. (The difference between 200mm and 300mm is not that great magnification wise.) Also, if you don't like you can sell it on eBay for about the same price as you buy it.
For Macro photography of the fish tank, I would seriously consider an inexpensive point and shoot (Nikon L1 maybe?) I know it sounds crazy, but you can get as close in and as sharp for $100 less than a macro lens. If you do buy a macro lens, I really like the 105mm macros from Nikon or Sigma. The sigma is less expensive and a little slower to focus. The benefit of this size macro lens is that you won't have to be IN the fish tank to get 1:1 reproduction. For ~50mm macro lenses (less expensive) maximum magnification is usually around 8" from the sensor plane (so you have to subtract the length of the lens.
Hope that Helps.
P.S. I own the Nikon D-70s and love it.