It's not the size; it's the shape. Specifically, it's the aspect ratio. That's the relative "squareness" of a photo format. For various historical reasons, there are a lot of different ones, and, as you've noticed, they don't line up. See What historic reasons are there for common aspect ratios? if you're interested in exactly why we ended up in this situation.

People selling prints in these different formats usually crop for each one manually. You can also "letterbox" — either with a border added digitally or (better, I think!) as physical matting within the frame.

Some print services offer a variety of aspect ratios in many different sizes. See for example this list of sizes from professionally-oriented Bay Photo. You can generally find something that fits the aspect ratio of your original in any approximate size. Others, and particularly budget online services, will just give a few standard choices (see for example mass-market print service Shutterfly).

If you're limited by your service's print options (or perhaps by wanting to fit in a standard off-the-shelf picture frame), one option is to order a size larger than what you want even when it doesn't match the aspect ratio of the original, leave a border, and trim the print at home.

If you plan to print a particular image in multiple formats, it makes sense to keep this in mind while shooting — leave space at the edges which you can adjust without harming the main point of the image. (With many artistic photos, this isn't possible, so you may need to take one of the other choices.)

Specifically, most DSLRs use a 2:3 aspect ratio — the long edge is 1.5× the short edge. That matches 4×6" prints, but not the next-size-up standard 5×7" — the long edge is slightly too short. If you're taking a portrait with the intent of printing at both sizes, you might make sure that there's extra headroom to be cropped off. This is even more so with the closer-to-square 8×10" prints often used for framed portraits.

Most point and shoot cameras, by the way, use 3:4, so there are no standard print sizes which match exactly! Both 4×6" and 5×7" prints are too small in the shorter direction, and so will need those edges trimmed off.

If you can shoot for this, cool. If not, use one of the workarounds.