This page describes how to join a Google group, using a non-gmail address. You might consider converting to use a gmail address, though, to get the benefits of the best free anti-spam solution in existence. But that is a different story, and if you establish a Google account with a not-gmail address, you can convert to gmail later (see below).
You should go to http://google.com, and be sure that you "Sign out" from that account, before following the rest of this process.
If you have a Google account, and wish to use it, it is possible to add multiple, non-gmail, email addresses to a Google account: but you cannot add gmail accounts, nor merge two Google accounts. So I recommend keeping separate Google accounts for each email address: gmail comes with its own Google account, and new Google accounts can be created for other email addresses if needed. It is a bit of a pain to have multiple Google accounts, because you can only be logged in to one Google account via one browser at a time... so I run multiple browsers (multiple copies of Firefox).
If you know the name of the group, but haven't been invited, you can go to http:/groups.google.com/group/NAME_OF_GROUP (substituting the real name of the group for "NAME_OF_GROUP", of course), and click on "Sign in and apply for membership". During the sign in process, if you haven't already established a Google group for the email address of interest, you may need to click "Create an account now", and fill out the form to establish a Google account. Once established, if you have lost the group signup context, you can start from the beginning of this section again.
Once signed in to the appropriate Google account, then you can request membership in the group. If it is a private group, you may not get accepted, especially if you heard about it from a friend of a friend. But public groups will generally accept you quickly.
If you have an invitation to join the group, and you wish to, then you should click the link in the invitation email that suggests it is the one to click to accept the invitation. You should not click the "abuse" link, unless you are getting repeated invitations for the same unwanted group.
Like above, during the sign in process, you may need to establish a Google account, if the email address you are using isn't already associated with a Google account.
Once signed in to a Google account, you should be quickly accepted as a member of the group. If, somehow, you have lost the context of your invitation, once you get signed in to your Google account, you can go back to your email, and click the link in the invitation again.
There is one, and only one, way to convert a non-gmail Google account to use gmail: sign in to that Google account, and then click the My Account link at the top, near the right. This should take you to your Personal Settings page. On this page, under "Try something new" is a link for Gmail. This is the appropriate starting point for adding gmail to a Google account that doesn't already have gmail. It is also the easiest way to convert from using a non-gmail account to a gmail account, if you wish to transition your email, rather than just starting with a new gmail address, and just ignoring the old one, or signing in to both for a while.