Show only current directory name (not full path) on bash prompt

Change the \w (lowercase) to \W (uppercase):

PS1='${debian_chroot:+($debian_chroot)}\[\033[01;32m\]\u@\h\[\033[00m\]:\[\033[01;34m\]\W\[\033[00m\]\$ '
                                                                                       ^^
           this one waaaaaay over here ------------------------------------------------+    

Have a look at the Bash Prompt HOWTO for lots of fun details. example:

user@host:/usr/local/bin$ echo $PS1
${debian_chroot:+($debian_chroot)}\[\033[01;31m\]\u@\h\[\033[00m\]:\[\033[01;36m\]\w\[\033[00m\]\$ 

user@host:/usr/local/bin$ export PS1='${debian_chroot:+($debian_chroot)}\[\033[01;31m\]\u@\h\[\033[00m\]:\[\033[01;36m\]\W\[\033[00m\]\$ '

user@host:bin$

The PROMPT_COMMAND variable, if set, is a command that gets run before displaying the prompt specified in PS1. In your case, PROMPT_COMMAND runs an echo statement with certain ANSI escape sequences that manipulate the titlebar of an Xterm.

If you suspect your PROMPT_COMMAND is overriding your PS1 prompt, you can unset it and test things out:

$ unset PROMPT_COMMAND

Finally, be sure that you're changing the PS1 definition that actually gets used. Common locations are /etc/bash.bashrc, /etc/profile, ~/.bashrc, ~/.bash_profile, ~/.profile. The system files are generally (but not always) run before the user files.

Simple bash replace command is

${VAR/pattern_to_find/pattern_to_replace}

For showing the last directory you can just do ${PWD/*\//}, i.e. find any thing before and including the last '/' and replace it with nothing.

On my ubuntu machine I use:

export PS1='$(whoami):${PWD/*\//}#'. 

My solution is to show the top three and bottom 2 directories when there are more than 5

So my prompt (which has other info too) looks like:

08:38:42 durrantm U2017 /home/durrantm/Dropbox/_/rails/everquote

when my pwd is actually

/home/durrantm/Dropbox/93_2016/work/code/ruby__rails/rails/everquote

My PS1 prompt is setup as follows:

HOST='\[\033[02;36m\]\h'; HOST=' '$HOST
TIME='\[\033[01;31m\]\t \[\033[01;32m\]'
LOCATION=' \[\033[01;34m\]`pwd | sed "s#\(/[^/]\{1,\}/[^/]\{1,\}/[^/]\{1,\}/\).*\(/[^/]\{1,\}/[^/]\{1,\}\)/\{0,1\}#\1_\2#g"`'
BRANCH=' \[\033[00;33m\]$(git_branch)\[\033[00m\]\n\$ '
PS1=$TIME$USER$HOST$LOCATION$BRANCH

git_branch is a function which shows current git branch, I keep it in my dotfiles, it is:

git_branch () {
  git branch 2> /dev/null | sed -e '/^[^*]/d' -e 's/* \(.*\)/\1/'
  }