I want to apply these 2 styles to 2 different categories: 

Category 1:

.wpfl-2 .specific-class-cat-1 .wpforo-category {
background: #fff!important;
}

Category 2: 

.wpfl-2 .specific-class-cat-2 .wpforo-category {
background: #000!important;
}

I'm exploring an option to add .specific-class-cat-1 or .specific-class-cat-2. Or if some forum-specific class already exists, what is it? 

Thanks, hope this helps clarify. 

Maybe I was not clear with my question. I am familiar with what you have described. What I'm after is to be able to style each forum or category differently. 

Example:

Forum "Business" - red color background

Forum "Leisure" - blue color background

Doing what you have described will style both forums "Business" and "Leisure" the same color. But I would like to style them differently. For that reason, I was looking to see if there is a way to add a forum-specific class or id to each forum and category? 

Thank you

Below is provided an example of how you can customize the categories. 

For example, if you need to change the background color of only the first category you should use this code: 

#wpforo #wpforo-wrap .wpfl-2:first-of-type .wpforo-category {
   background: #d8b3b3 !important;
}

You can find all classes by pressing F12 on the browser and opening Chrome or Firefox Console:

(And hadn't worked out how to do private messages on this site either, not something my site really needs.)

OK, will send a link in a bit, though being realistic it's a nice to have, rather than an essential.

We are not able to access your forum we see the message below even if we logged in via the details you've provided. 

Maintenance mode

The site will be available soon!

I'm keen to do this as well - but my forum is private so can't post the URL?

You can create a new private topic and leave your forum URL. 

(Had just come to post a query about what the "Forum color" option does in Forums>Forums>Forums and Categories>Additional Options does? As it looks like that should change the forum (or category) colour but doesn't appear to?)

The option changes the icons of the forum in the threaded layout. See the screenshot below. 

please send the login details with your forum URL to infogvectors.com email address. 

(Had just come to post a query about what the "Forum color" option does in Forums>Forums>Forums and Categories>Additional Options does? As it looks like that should change the forum (or category) colour but doesn't appear to?)