How to integrate wordpress?

I’m totally new to wordpress and have just installed it.

Then start the learning process:
Main_Page#Getting_Started_with_WordPress

I am hoping to use it as a means of different people being able to write articles

That’s possible, of course. Study the Users section of your admin panel.

which will then get published in certain pages on my existing site.

That’s not possible.

I’m wondering if there is some code I can use to get the latest post from a category and displey it on my page.

Yes, there might be, but you need to learn more and/or give a more detailed description of what you want.

As a general note. WP is not something to integrate – it is a stand alone blogging tool (maybe even CMS)… so it is possible that going the other way around, i.e. integrating/managing your site in WP would be more beneficial on long term.