Just READ that you are using UBB Classic.
Thus the particulars of what I said do not apply.
Sorry to hear that the admin of your community passed away.
To answer your question:
It all depends on how registrations are set up. There is no action required on your part if e-mail verification is being used.
From what you say, it sounds like the board is set up to notify the admin of new user registrations. Read the e-mail. What does it say?
Does it contain "There is no approval required, but you have chosen to be notified of all new user sign-ups."?
If so, it is all on the new user to click the link in the e-mail sent to them. It is also on the new user to actually read the instructions during the registration process.
You will want to check how the previous admin had registrations set up. In the CP (Control Panel), go to Registration Settings>Basic Options Tab.
Likely the third check box "Enable Email Verification?" is checked. If so, then what I said above applies.
Once a new user registers, the forum will send an e-mail to the address given during registration. The new user then has 24 hours to check their e-mail and click the link contained within. During this time, if you go to the CP opening page, you will see that there is a user in the "Registration Queue". If the 24 hours elapses and they do not take the required action, the account is automatically deleted as if they never did anything.
From time to time I get an e-mail from someone who registered and then did not click the link in the e-mail that was sent to them. They rag on me for not granting them access. I reply back and politely point out that they did not read the instructions and that they will need to register again and to READ the instructions.
Another option is that the "Enable Registration Queue?" box is checked. If so then ALL new users are place in a queue and you need to approve them. All you need to do is go to the CP main page and you will be notified if there are any new registrations to be approved. Just click on the link and you will be taken to the page where you can approve them (or not).
I suggest you create a new user for yourself and see how this process works both from the perspective of a new user and Admin.
It is also a good idea to have a normal user account on your forums so that you can test things out from a user perspective.
EDIT: One last thing is that you will go nuts if you try to find where in the CP you can select (or not) to have the forums notify you via e-mail of new user registrations. That's because it's not there.
You need to be logged into the forums (as opposed to the CP) and go to My Stuff>Edit Preferences.
Scroll down to the bottom and find the heading "Moderator & Administrator Preferences".
Non-intuitive? You bet.
If you are going to take over caring for your community, there will be a bit of a learning curve, but you will catch on. Good luck.