TL;DR: Download JDK 1.6, then point a Gephi startup file to it maybe with one change (/Library... and not /System/Library...).
The longer version is that Gephi, which I love, hasn't had an update (as of this writing) in quite some time (parts of it have, parts haven't) and it doesn't work with the newer versions of Java. The 0.9 version download file for OS X doesn't seem to be where the link says it should be (again, as of this writing in early October, 2015).
Apparently Gephi 0.8 needs Java version 6, not version 7 or 8. Java appears to have lots of parts, names, and maybe even v6 is "1.6" and v7 is "1.7" which I don't understand but I don't do Java so I'm not going to spend time figuring it out--I got Gephi to work, that was all I cared about right now.
My new laptop, to replace my five-year old one, didn't even have Java on it (ah, the purity of it all). I got the JDK (Java Developer's Kit) 1.6 from part of the Apple Support website, which is what you want.
But you have to tell Gephi about it, as far as I know. This article was awesome except it turned out on my machine not quite right (but close enough that I figured it out). Java wasn't where I thought it would be. This article on Stackoverflow helped me find Java 1.6, although note the article is about Java 1.7, so make the command this:
/usr/libexec/java_home -v 1.6
(Ok now you don't need to read the Stackoverflow page, just run that in your terminal if you've installed 1.6 and don't know where it is.)
Turns out, I don't know why (and I don't care, it's working) Java 1.6 wasn't in /System/Library... it was instead in the same path but not the one in /System, it was just in /Library....
So, put that line in the Gephi startup file near the top as per the directions (I made it the first command) and so far Gephi actually loads, which it wasn't doing before. Granted I haven't actually tried to do anything with it, and who knows if this will blow up other Java things (or maybe my machine really didn't have any Java on it at all).