The constraint that defines it
In SDR the brightest value the encoding can carry is diffuse white: the brightness of a sheet of white paper under the scene's lighting. There is nothing above it. A specular reflection off chrome, a sunlit window behind an interior, a candle flame, all of them land at the same ceiling as the paper and render as the same flat white.
This is not a bit-depth problem and more bits will not fix it. It is what the range means.
What HDR changes
HDR keeps diffuse white roughly where it was and adds room above it. That is why converted footage does not simply look brighter: the parts of the frame that were already correct stay put, and the highlights that were flattened get somewhere to go. See what actually changes when you convert.
Identifying an SDR file
ffprobe -v error -select_streams v:0 \
-show_entries stream=color_transfer,pix_fmt -of default=nw=1 input.mp4
color_transfer=bt709 (or an unset field, which usually means the same) with pix_fmt=yuv420p is SDR. Anything reporting smpte2084 or arib-std-b67 is not.
A file being SDR has nothing to do with how bright it looks on your monitor. A modern display shows SDR content stretched across its full brightness, which is why a flat file can look punchy and still carry no highlight information at all.
Questions people ask
Is SDR bad?
No, it is a constraint that most content was made within and looks correct inside. SDR grading is a mature craft and a well-graded SDR file beats a badly converted HDR one. The limit is specific: nothing can be brighter than diffuse white.
Should everything be converted to HDR?
No. Footage whose subject has no light source above diffuse white gains little, and screen recordings gain almost nothing. Convert the clips with a window, a flame, a reflection or a sky in them.
Related: how to convert an SDR video to HDR.