I need to get all file names for a directory
for file in "$1"/*;do
    echo $file
done 
gives me directory_name/file1_name, directory_name/file2_name.
However, I just want the file1_name
You can use parameter expansion to get only the filename before printing as follows:
    for file in "$1"/*;do
        file="${file##*/}"
        echo $file
    done
You can find more about parameter expansion over here : Shell Parameter Expansion (Bash Reference Manual)
