You get 20 articles a month, unless you're following a link from Google, social media like Facebook or Twitter, or from blogs, in which case you can view the linked article for free.
If you want to view more than 20 articles a month, you can buy a subscription. Subscriptions include unlimited new articles and up to 100 archived articles. More archived articles or access to premium online crossword puzzles are available for an additional fee. Oh, and as far as I'm aware, even with a subscription your NYT digital experience will still have a boatload of big, annoying, animated display ads.
Anyway, how much does this subscription thingy cost? If you want to use the NYT's website and one of their smartphone apps, it's $15 every four weeks (not per month; you don't get any free weeks from these guys). If it's the website and a tablet app you want you want to use, it's $20. If you want to experience the Times on every possible digital device, it's $35.
If all of that is too confusing, note that print subscribers get unlimited digital access for free. Here in Chicago a 7-day weekly print subscription is around $5 cheaper than their "unlimited" digital subscription, and includes the dead-tree edition of the paper.
Or you can just Google the title of any article you want to read, and read it for free.
I'm not saying the Times shouldn't make money, but would it have killed them to come up with a product simple enough for a typical consumer to understand?