So you want to copy all the .html files in some source directory and its subdirectories, all to a single directory (i.e. collapsing the hierarchy)?
POSIX Standard:
find myDir -name '*.html' -type f -exec sh -c 'cp "$@" "$0"' ~/otherDir {} +
Note that ~/otherDir becomes parameter 0 to the intermediate shell, which allows for the source files to be precisely "$@". Leaving the target directory outside the shell has the additional advantage that you won't run into quoting issues if that's a variable in the parent shell script (-exec sh -c 'cp "$@" "$0"' "$target").
For older systems that don't have find … -exec … +:
find myDir -name '*.html' -type f -exec cp {} ~/otherDir \;
I your shell is bash ≥4 or zsh:
shopt -s globstar # only for bash, put it in your `.bashrc`
cp myDir/**/*.html ~/otherDir/