Why when i enlarge my photo it gets cut off?

I just purchased a Nikon D3000, and have been taking photo's. However, when I enlarge a photo from a 4x6 to 8x10 and print it; I've notice that the subject gets cut off.

That's because the aspect ratio is different. You are shooting in an aspect ratio or 3:2 so it can produce a 4x6 print because that's also a 3:2 (or actually a 2:3) ratio.

8 x10 (A4) is not a 3:2 ratio - you need a paper size of 8 x12 to get a 3:2 ratio. Standard paper sizes (A4, A3) are not in the same ratio as photographic prints are.

You need to frame your subject so that you can crop it to an A4 size after shooting - then you'll be able to print on A4 paper.

A 4x6 photo is a 3:2 (1.5x factor) ratio and an 8x10 is a 5:4 (1.25x factor). An 8x10 is about the closest you can get to a square ratio in STANDARD print sizes.

When you crop a photo to a certain size there will almost always be part of the image that gets cut out.
If you want to enlarge a 4x6, the next size up with no cropping is a 20x30. With an 8x10 it would be 16x20.

When I print photos, I take the full image and then crop it to the aspect size I want to print.