Last night I was practicing adding users using the passwd and shadow file. Used the passwd command to create a password for the new user. Made a home directory for the new user. Copied the default login initialization files to the users home directory.Assigned the appropriate permissions for the home directory and contents.Set the owner and group for these directories. However, when I attempted to login as that user I got the following strange looking prompt.
bash-2.05$. Can anyone tell me why that is and how it can be changed?

i'd guess that, assuming you DID copy .bashc and .bash_profile from /etc/skel or another user, that they do not have the correct access rights, and consequenetly the global /etc/bashrc file which would normally set the prompt, path etc could not be accessed. try chmod-ing them to the right name, assuming they exist in the first place. same might be required for the directory you created.

the "useradd" command is there for a reason...

The commands I used included:
#cp /etc/probile /home/user/.profile
#cp /etc/bashrc /home/user/.bashrc

I replaced the user option (or should I write variable) with the username of the actual login name of the user I created. Is there anything wrong in either one of the 2 above commands?

Yes both of those commands won't work.

Both /etc/profile and /etc/bashrc are global files. They are used to set things for each user on the system. The .profile and .bashrc in the users home directory are intended for any extra changes the user wants to make which are specific to his/her needs.
So by doing what you did everything in profile and bashrc will be run twice for each user.
Also if you where root when copying the files then the owner will still be root. Those files should be owned by the user itself. So you'll have to run another command like: chown user.user /home/user/.profile

The prompt you got is the default bash prompt usually you would add a line in .bashrc something like the following:
export PS1='\u:\w\$ '

You definitely don't want to copy these files from /etc. As Acid suggested, just copy the files out of /etc/skel, change permissions and you should be set.

When I did this yesterday I noticed that I could not access a number of commands including shutdown. Did I inadvertently change root's own profile? If so, how do I correct such an error?