Biblical Christianity vs. un-Biblical Christianity
In regards to protestantism, Biblical Christianity could probably be defined or characterized by how one would describe un-Biblical Christianity. What is unbiblical Christianity? It is Christianity (or the claim thereof) that lives or teaches something in contradiction to Scripture.
Naturally, to conclude that something contradicts the truth of the Scripture, one must have an idea of what is meant (ultimately, by God) by what is written. That involves fallible humans, and therein is a source of inconsistency and disunity.
In general, Biblical Christianity means that you live and believe things consistent with the testimony of Scripture. Though some might claim something as narrow as @Anonymous ("whatever-I-believe-in"), I don't think this is what is meant by most people who use the phrase. They are, at least, claiming that what they believe is not their own idea, but is taken from the Bible.
A related phrase, "that's [not] Biblical," answers the question as to whether or not an idea is compatible, harmonious, or incompatible with the Scripture. For people who claim to hold to sola scriptura, the Bible is the ultimate check of any idea, and so everything a Christian believes and does should "be Biblical" because those things should all be harmonious (or at least compatible—an allowance for people who don't agree with you but whose conclusions are at least a reasonable interpretation) with the testimony of Scripture.