I kind of want to get the intel xeon phi co-processor since there is a model which seems to be running for $230. I have two questions. Can I fully utilize the capabilities of this just using gcc along with openmp or will I need the intel compiler. Also what is it about this model which makes it so cheap?
3100 series is a first generation of Xeon Phi (codenamed Knights Corner, abbreviated KNC).
Using GCC for Xeon Phi KNC programming is definitely not perfect idea. See for example: Xeon Phi Knights Corner intrinsics with GCC
So it's extremely recommended to use Intel Compiler for KNC. And yes, in case of non-commerical use, you can apply for free Intel Compilers license here: https://software.intel.com/en-us/qualify-for-free-software (this is kind of new program, unavailable in past).
Given KNC price tag is low enough, although I periodically observe KNC sales for similar prices (so at least it's not "incomplete" Phi; and it's not cheating, although Gilles' passive cooling point is valid). I don't know which problems you work on, but you should be aware that KNC is most of all suitable for some highly parallel workloads. There is a good reference of types of applications which could benefit from using Xeon Phi KNC: https://software.intel.com/en-us/articles/intel-xeon-phi-coprocessor-applications-and-solutions-catalog
As I mentioned in the beginning, you are asking about first generation Xeon Phi. Many things (including GCC answer) will likely change with introduction of second generation of Xeon Phi (codenamed Knights Landing, KNL) to be publically released in ~next year.