My elderly mother purchased an Apple computer a few days ago and we just purchased a new all-in-one printer-scanner-copier. When installing I get stopped requiring the password and she can't remember it. How do we get around it or reset it?

There are methods to reset the password via the command line you can Google "change lost OSX password" however You will need to backup the home folder as described below to be safe. For most users I would suggest reinstalling unless you have previous Unix, Mac or Linux command line experience.
The user section of System preferences is password protected so unless she remembers the password or there is a second admin user who can login she can not change the password. If Auto login is enabled drag the users home folder to a USB flash drive (you can check the folder size by highlighting it and pressing 'Command' and 'i' to make sure you have enough space the active user's home folder will appear with a house icon. Then use the disk supplied with the computer to reinstall the system adding back the user remember to set the "short name" to the same name of the user folder you backed up. You can then drag the contents of the backed up folder into the new user folder. Do not replace the "Libraries" folder because some items may cause some problems. Run your updates and make a second user with administrator rights. Store both passwords in a safe place. The second account will be useful when you need to trouble shoot problems.
Mac uses Sudo to give users admin rights temporally this is much more secure than Windows but dose require that you enter your password to do many higher level functions like installing software and editing users. Your login password is your Sudo password.
This service should be offered for a fee by a Mac Certified technician. Or your Apple store