In the 1940s, Cornelius Van Til and other professors at Westminster Theological Seminary first tried to block Gordon Clark's entry into the Orthodox Presbyterian Church, then later put him on trial for removal from the denomination. What theological issues were at stake in the controversy, why was it such a serious issue that Clark was removed from the OPC, and who are today's Clarkians and Van Tillians? For the last question, I'm not looking for a list but lines of demarcation.
To get it straight from the horses' mouths, you can read Van Til's complaint [PDF] and Clark's answer [PDF] online. Both are very long and in-depth and also spend a good deal of time on procedural matters before getting to the theological. But if you really want to know and understand the ins and outs of what was at stake, primary sources are usually the best way to go.
With that being said, the paper The Gordon Clark and Cornelius Van Til Controversy by Jared Moore [PDF/MP3] gives a good distillation of the controversy:
Clark believed that man's knowledge and God's knowledge are quantitatively different but not qualitatively different. Instead of a two-fold theory of truth, he believed that truth is one. If man knows an item of truth, and both God and man know the identical item, then on this item God's knowledge and man's knowledge coincide. Yet, man can never know exhaustively and completely God's knowledge of any truth in all its relationships and implications because man is not omniscient. Every truth has an infinite number of relationships and implications and these implications in turn have other infinite implications. Thus, man's knowledge will forever remain quantitatively less than God's knowledge, even in Heaven.
Furthermore, Clark believed that God is logic. His paraphrase of John : was, "In the beginning was Logic, and Logic was with God, and Logic was God. . . In Logic was life and the life was the light of men." To Clark, "God and logic are one and the same first principle, for John wrote that Logic was God." Thus, he held that Scripture is a part of God's mind. And if God has spoken in Scripture, He has spoken logically in logical organization. Since man was made in God's image, he can receive this logical knowledge of God because "human reason is not so much human as divine." Clark believed that truth is the same for God and man. He wrote, "Naturally, we may not know the truth about some matters. But if we know anything at all, what we must know must be identical with what God knows. God knows all truth, and unless we know something God knows, our ideas are untrue. It is absolutely essential therefore to insist that there is an area of coincidence between God's mind and our mind." Thus, Clark believed that Van Til could never say he knows truth since he never knows anything as God knows it.
Van Til, on the other hand, believed that there is a Creator/creature distinction between God and man that makes God quantitatively and qualitatively incomprehensible to man. He argued that God's knowledge of Himself is archetype knowledge. God also has an analogical knowledge of His archetype knowledge, which is God's ectype knowledge. God has given man in His revelation, His Scripture, analogical knowledge of His ectype knowledge. Scripture is the ectype of the ectype knowledge of the archetype knowledge of God. Thus, man knows nothing of God's archetype knowledge since he is not God. Man can never know anything as Creator; man can only know as creature. Nevertheless, because man's knowledge of God as revealed in Scripture is analogous to or an interpretation of God's ectype knowledge—which is analogous to or an interpretation of God's archetype knowledge—Scripture is God's truth revealed to man. Scripture is an accurate interpretation of God's knowledge because God is the Interpreter. However, man's knowledge of Scripture is always a creaturely knowledge. God's knowledge and man's knowledge are always qualitatively and quantitatively different. All revelation is anthropomorphic. If man knows as God knows, then man is God. Thus, Van Til saw a fatal flaw in Clark's doctrine of the incomprehensibility of God.
The issues therefore had to do with epistemology, God's incomprehensibility, and the creator-creature distinction. In addition to the documents I already linked, Moore's paper cites Clark's essay "The Axiom of Revelation," which is included in the anthology The Philosophy of Gordon H. Clark, and Van Til's books Defense of the Faith and Introduction to Systematic Theology as case studies of their overarching philosophies.
Moore's paper has a long section on historical narrative of the controversy, which should prove useful if you want to delve deeper. I'm told Herman Hoeksema's book The Clark-Van Til Controversy is also a good read on the topic, but Hoeksema is said to have a detectable pro-Clark bias throughout (but probably not as pronounced as Moore's pro-Van Til bias). Interestingly, Hoeksema's opponent in the common grace controversy, Abraham Kuyper, was one of the formative influences on Van Til.
Why did it matter so much?
As for why these issues were a big enough deal to justify driving Clark out of the OPC, Moore's paper has an answer for that too:
The implications of Clark's theology undermined the Reformed Faith when taken to their consistent end. When the Creator/creature distinction is compromised, and man believes he can eliminate Scriptural paradoxes and comprehend all Scripture logically, the consistent end is a denial of the Christology expressed in the Chalcedonian Creed and the Westminster Confession. ... Clark's eventual rejection of God the Son's single personhood justified Van Til's original concerns, for if Clark's conclusions were taken to their consistent end in the OPC, the result would be a denial of orthodox Christology.
Specifically, Clark apparently ended up confessing that Christ subsisted in two persons, though there are many who defend Clark's orthodoxy even on this point, claiming that he was defining "person" differently from the historic confessions, and that his work can be read as consistent with them.
The Clark-Van Til divide today
These days, Clark and Van Til are both widely respected in Presbyterian and reformed circles, both as theologians and as presuppositional apologists. But one of the lasting legacies of the controversy is that their brands of presuppositionalism are recognized as different: see Van Tillian presuppositionalism versus Clarkian presuppositionalism.
After Clark's exit from the OPC, he joined a denomination that, many mergers later, became the Reformed Presbyterian Church Evangelical Synod. In 1982, the RPCES merged with the Presbyterian Church in America. Partly because a merger between the PCA and OPC appeared to be on the horizon (though it never came to fruition), Clark refused to join the PCA and joined the breakaway Covenant Presbytery (which later called itself the Reformed Presbyterian Church in the United States) in 1983.
Those in the OPC tend to be Van Tillians, unsurprisingly, but from what I understand, denomination isn't otherwise a reliable indicator of support for Clark or Van Til. John Frame and Francis Schaeffer are two of the well-known theologians who have been heavily influenced by Van Til. For his part, Frame said that "the controversy was really unnecessary and largely based on misunderstanding. Van Til in my view was at his worst when he was debating with other Christian apologists." On the other side, John Robbins' Trinity Foundation consciously and proudly stands in the Clarkian tradition.