The New York Times follows very strict typographic rules. They use a total of two typefaces in their entire publication (barring any ads that might use different ones). The two typefaces are Cheltenham (which is used for Headlines, deks, subtitles, and cutlines) and Georgia (used for body copy).
Headlines, Deks, Subtitles
Instead of using different typefaces to distinguish between the three different lines of text, the New York Times changes the size, weight and capitalization to get across each lines importance.
The headline is all uppercase, bold, and the largest type.
The subtitle is still all uppercase and bold, but they made the type smaller to make it less important.
Finally, the dek is still bold, but it is not uppercase and it is the smallest size of the three lines.
These are not hard and fast rules though. The New York Times uses different weights and decorations (italics, uppercase, etc.) on different headlines to make the more important stories stick out.
Here are examples of different headlines all from the same paper:
The cutlines are Cheltenham type as well, but they are set in the smallest typeface and are bold, almost the same size as the body copy, which makes them very hard to distinguish from one another (because Cheltenham and Georgia are very similar typefaces).
Bylines & Body Copy
The Bylines in the New York Times are set in Georgia and are bold and uppercase to set them apart from the body copy, which is set in Georgia and is the smallest type on the page. The body copy is much more dense compared to the rest of the type on the page.
Overall, I think that the New York Times minimalist outlook on typography (only using two typefaces) is meant to get across a sense of sophistication and dependability. Both type faces they use are serif, which make them seem more professional and wise (if you will). And by depending on size and weight to differentiate type rather than using different typefaces altogether makes the newspaper look simple and consistent for readers.