How can I tell which of my pages are single-view pages?

First off, take a look at Conditional Tags in general.

I am trying to understand what the is_singular() function does.

Straight from the codex: “This conditional tag checks if a singular post is being displayed, which is the case when one of the following returns true: is_single(), is_page() or is_attachment().

What I really need to do is determine if a page is a category page.

The conditional tag that applies here would be is_category(), which returns true, if a category archive page is displayed.

But the problem is that I am not sure how to determine which are single-view pages, regular pages, and category pages.

As you do not care about attachment pages, is_singular() obviously isn’t for you.
Still something can be learned from its description – namely, the other two tags (the category one aside) needed to determine whether your condition is met:

is_single() and is_page()

Hence,

if ( is_single() || is_page() || is_category() ) {
    // do your thing
}

is what you want.

Should chaining of expressions in PHP be new to you, take a look at its logical operators.