Suppose we have a 806x600 pixel image, which just happens to have been scanned at 750 ppi from a 35 mm color negative. It is to be printed at 240 ppi to give about 3X film size, or 3.36 x 2.5 inches, to fit the space on the page
A serious technicality is that inches cannot be shown here on the video screen (where pixels are displayed directly), so this copy is smaller, of some arbitrary size. The example however is about printing a 806x600 pixel image to be 3.36x2.5 inches on paper.
Adobe Elements and Photoshop have the same resize dialog box, and their menu Image - Resize - Image Size is shown at left. Newer versions of Paint Shop Pro and PhotoImpact are very similar, and Paint Shop Pro is shown at bottom. The Resample checkbox chooses Resample or Scaling mode, so if scaling, the first thing you do is to uncheck Resample Image.
The top area shows the image size in pixels. You can change image size there when resampling but this area is disabled when scaling (as shown here), so that the pixel dimensions cannot be changed. The main idea of scaling is to NOT change the number of pixels in any way. Constrain Proportions (aspect ratio) is also disabled, because any change would require resampling pixels. The lower area shows the dimensions of how the flower image is scaled to print on paper.
Note the main point again. The top part of the dialog which shows pixel dimensions is disabled (grayed out) when we do not specify the Resample checkbox. Then we are scaling, meaning we cannot change the pixel dimensions then. We only specify the printed size on paper then - some number of pixels at some value of pixels per inch will cover some inches of paper. But no pixel is changed.
To preserve the original aspect ratio (shape of the image), then Constrain Proportions means that when you retype one new dimension in inches, the other dimension and resolution will automatically change as necessary to scale this image to that printed size. Or if you enter the printing resolution you want, the inches will track accordingly. Experiment a bit, you will easily see how it works.
As long as Resample Image is not selected, we are not resampling, and the file size will remain at the original size. This introduces no quality loss because the original pixels are entirely unaffected. You will not see any change until you print it (with File - Print), but it will print at this new size and resolution. Corel Photo-Paint and Picture Publisher call this scaling selection Maintain File Size. PhotoDeluxe calls it Constrain File Size. Adobe unchecks the Resampling checkbox. But it is all the same idea, they work the same - Constrain or Maintain Size means Scaling (Resample is disabled). The idea of scaling is that the pixel dimensions do not change, therefore the file size does not change either.
If you do want to resample to a different image size in pixels, then obviously you cannot "maintain file size". Select Resample mode to enable the pixel dimension fields so you can enter any new pixel dimension directly. If more convenient, you can change the units to Percent (below) to specify perhaps 50% for either dimension, which will calculate 403x300 pixels here. Or in resampling mode, you may enter a new size in inches, or a new resolution, and it will resample to create the required number of pixels to fit that specification. If you uncheck Constrain Proportion, you can change width and height individually, but that shape distortion is normally unacceptable for photo images.