Person logins

Firstly they must be person under Contacts > People. If this is a new person, just click Add. If they are already there, just click Edit.
Ensure they have a Username set. This can be anything but must be unique. Using the person's email address is a good option.
For security you can't set a password for them. A password should only ever be known to them. So under Password, tick Send email with instructions on how to choose their password.
Ensure Status is set to Active.

Things to check:

Do they have a Username set under their profile?
Are they using their Username? Sometimes people enter their email address when the username is actually something else.
Is their Status set to Active? If it's set to Disabled, they won't be able to login.
Are they using the right password? Perhaps get them to do a password reset. Remember, passwords are case-sensitive.
Do you have a login link for them to click on? (see below).

There's a few ways a person can login:

By clicking on the login link in the utility navigation at the top of the page. When a site is first created we add this by default. If you have an existing site, or have removed it in the past, you can re-add it. Just go to Website > Settings > Navigation, under Utility navigation links click Add, change the Type to Login, then click Add.
If any page is set to require a login to view it, then we always present the person with a login screen if they aren't already logged in.
On the checkout page, there's a login button they can click,
They can always access the login directly by typing /account/login after your domain name in the address bar.

You can optionally require people to be logged in before they can view pages or products.

Find the page or product and click Edit.
Choose the Person tag required to view. Only people with this person tag can view the page after logging in.
Click Update.