Better read this (and alike) first:
http://en.wikipedia.org/wiki/I686
x86_64 means a 64-bit distribution that will only install and run on a 64-bit machine (you know, some cpus sold nowadays are 64-bit, most are 32-bit). Among 32-bit machines there are various flavours, that are marked with i*86 (i386, 486, 586, 686). It won't hurt your machine to run a "i386 Fedora", though your 'uname -a' states your machine is "i686"
Read more about this at Wikipedia (or any other website) and it'll become more clear to you. Shortly said, almost every pc can run i386-marked software, but only some (new enough) can run i686 for example.
If something might hurt your machine, you would probably be informed about it.