get the name of the file

basename $path

from relative to absolute

MYPATH="../whatever"
MYPATH=$(readlink -f ${MYPATH})

is Absolute

if [[ "$PATH" = /* ]]; then
echo "absolute path";
else 
echo "relative path";
fi

case $PATH in
  /*) echo "absolute path" ;;
  *) echo "relative path" ;;
esac

from Windows to Linux and vice-versa

cygpath -u path # from Windows to Linux
cygpath path # from Linux to Windows

File or Directory

if [[ -d ${ARG} ]]; then
    echo "${ARG} is a directory";
elif [[ -f ${ARG} ]]; then
    echo "${ARG} is a file";
else
    echo "${ARG} is not a path";
    exit 1;
fi