"Media" is the plural of "medium." A "medium" is a means of communication, or even "exchange." If you use a river to transport logs into the city, the river can be said to to be a medium.
More broadly, we generally use "Media" to mean all of the communications tools used by news organizations and even quasi-news organizations . . . all of the means of communications. That means that "the media" refers to newspapers, magazines, television, radio and even the Internet.
In this case I believe they mean any media that is non commercial in nature meaning no wizzley, nothing where you have any monetization surrounding the picture anywhere. I'd have to see the site or their tos to really nail down what their pure definition of media is- perhaps they mean only free to use if your in the news media scenario, i.e. for use in news reports or only for use in video production. The definition of media is highly ambiguous so it's very hard to nail down exactly what they mean- but again, look through their tos, ask in the forum of the site or email the site owner (use whois to find the number or email addy).
What site is it?
edit:
I took AKC and Google'd it, landing on the american kennel club (is this it?) this seems to have dog shows etc.. and that tells me that the media in question here is not the general media but the media surrounding the sites purpose i.e. if your a reporter doing a story on dog shows or a specific dog show they had, they could write the story and use the picture if they attribute the site- otherwise a normal joe wanting to use the picture in an article NOT RELATED to or specifically about their site and mission- it's not allowed. In other words only news, newspapers, magazines, and sites reporting about the site, event, specifically can use them- you can't put them on wizzley because they mention no commercial use or otherwise... So unless you have a press pass for the show sorry...