For files in directory, only echo filename (no path)

If you want a native bash solution

for file in /home/user/*; do
  echo "${file##*/}"
done

The above uses Parameter Expansion which is native to the shell and does not require a call to an external binary such as basename

However, might I suggest just using find

find /home/user -type f -printf "%f\n"

Just use basename:

echo `basename "$filename"`

The quotes are needed in case $filename contains e.g. spaces.