How to hide computer name and user name in terminal command prompt
Change your prompt in your ~/.bashrc file. The example you asked for would be:
export PS1=&quot;\W \$&quot;

It would result in the current folder you're in being shown plus a $ for the regular prompt and a # if you're root. Check out this guide for more examples of what you could show in your prompt.
Edit:
As per one of the comments below, you might need to source your ~/.bashrc from your ~/.bash_profile or even put this code in your ~/.bash_profile instead. You can read this article for a better explanation on which file to use.