Homebrew requires macOS 10.15 or higher, while versions 10.10–10.14 are supported on a best-effort basis.
The one-liner installation method found on the Homebrew main page:
/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/install.sh)"
requires the Bourne-again shell, i.e., bash.
Note that zsh, fish, tcsh and csh will not work. (Without bash, you may get a non-descriptive error such as ”Illegal variable name.”)
So switch to /bin/bash (in System Preferences --> Users & Groups --> Right click on your username (must be admin) --> Advanced Options... or in the Terminal preferences) for the command above to succeed.