How do most forums make their money if affiliate links are not allowed? Is it only through Google ads and such? I just created a forum website and will list the best deals, steals, freebies, coupons, e-discounts, & penny pinching tips & secrets. How do you advise I best monetize on my new forum?

Who says affiliate links are not allowed? It's your forum, so you call the shots.

You can edit the forum to add in some banners to rotate to your visitors. Also, you can customize all of the e-mails that go to your members when they join, notifications of replies to certain threads, etc. to promote your affiliate links.

Also, whenever you post a new deal yourself, just include your affiliate links. You can also send out a weekly e-mail, generated through your forum (phpBB in your case) that highlights the best deals with your affiliate links.

phpBB also enables you to promote your forum via RSS feeds. Get that going, and you'll have even more distribution of your affiliate links.