They are taking a 'shotgun' approach to referencing the font. The browser will attempt to match each font name with any installed fonts on the user's machine (in the order they have been listed).
In your example "HelveticaNeue-Light" will be tried first, if this font variant is unavailable the browser will try "Helvetica Neue Light" and finally "Helvetica Neue".
As far as I'm aware "Helvetica Neue" isn't considered a 'web safe font', which means you won't be able to rely on it being installed for your entire user base. It is quite common to define "serif" or "sans-serif" as a final default position.
In order to use fonts which aren't 'web safe' you'll need to use a technique known as font embedding. Embedded fonts do not need to be installed on a user's computer, instead they are downloaded as part of the page. Be aware this increases the overall payload (just like an image does) and can have an impact on page load times.
A great resource for free fonts with open-source licenses is Google Fonts. (You should still check individual licenses before using them.) Each font has a download link with instructions on how to embed them in your website.