WHAT DENOMINATIONALISM IS
Congregational church government puts forward a system where individual congregations are jurisdictionally independent from other congregations. Church power lies only in the local congregation, and there is no power invested in church courts (presbyteries, synods) beyond the local congregation. Presbyterian church government, on the other hand, asserts that individual congregations are not jurisdictionally independent, but rather that the power of elders in the church is collegial, so that church courts made up of the leadership of multiple congregations can exercise church power over individual congregations.
However, many Reformed people today seem to favor a third model of church government--a position I will call denominationalism (or, alternatively, semi-congregationalism or semi-presbyterianism). Denominationalism is a middle ground position between congregationalism and presbyterianism. Like presbyterianism, it holds that individual congregations are not independent but are subject to the jurisdiction of higher courts made up of the leadership of multiple congregations. However, unlike presbyterianism, denominationalism holds that this jurisdictional accountability to higher courts goes on only to a limited degree. Such accountability does not exist between all churches, but only between some churches. Bascially, it works like this: A group of congregations in a region are united under mutually-binding higher councils (usually called presbyteries). These presbyteries are united under mutually-binding higher councils (usually called synods). Finally, synods are united under a supreme mutually-binding council (usually called a general assembly). (Sometimes the synod step is skipped.) So far this position is similar to presbyterianism. But here's where the difference is: In presbyterianism, all visible churches are to be united together under these mutually-binding councils, so that the highest court, the general assembly, would be a council representing all the churches of Christ in the world. But in denominationalism, there are multiple general assemblies, each of which represents a distinct collection of individual churches. There is jurisdictional accountability between the churches within each collection of churches, but none between the different collections. These collections, or groupings, of united churches are called denominations.
Consider these two denominations: the OPC (the Orthodox Presbyterian Church) and the PCA (the Presbyterian Church in America). Within the OPC, for example, there are a number of congregations which are united together under mutually-binding higher councils. Groups of congregations in a region are united under common presbyteries, and all the presbyteries are united in a common general assembly. Within the PCA, it is the same way. However, there are no common councils that bind together the general assemblies of the OPC and the PCA. These denominations are jurisdictionally independent of each other. Thus, what we have is something between presbyterianism and congregationalism. Instead of each congregation being independent from all others (as in pure congregationalism), we have distinct groups or clumps of churches. Churches within each "clump" (that is, each denomination) are united in a presbyterian manner, while the clumps themselves are independent of each of the other clumps (following the independency favored by congregationalism).
Now, the PCA and the OPC both officially embrace presbyterianism and not denominationalism. However, many in the ranks of both denominations consider the officers and church courts of both denominations to possess de jure legitimacy and authority. Now, in a presbyterian system, if both the OPC and the PCA possess de jure legitimacy, they would be required to function together under a common supreme council. The idea that the OPC and the PCA both possess legitimate church authority and yet are authorized to remain separate from each other is denominationalist in nature rather than presbyterian, and yet this is how many professed presbyterians actually think about things.
WHY YOU SHOULD BE A DENOMINATIONALIST
Denominationalist church government has a major advantage over both presbyterianism and congregationalism. Both of the other two forms of government are consistent. That is, there are principles at the heart of them that make coherent sense (whether they are correct or not). The principles underlying presbyterianism are that there is one Body of Christ and that therefore churches should not function independently, and that church power is collegial so that all congregations share in jurisdiction over the whole Body of Christ. The principle underlying congregationalism is that individual congregations are autonomous and that it is an unjust imposition for any congregation to try to exercise any authority over other congregations. Denominationalism, on the other hand, has no coherent or consistent principles. It shares elements of both the other systems but applies them completely arbitrarily. And this is good, for, as Ralph Waldo Emerson famously put it, "consistency is the hobgoblin of little minds." It is better to have a system that goes beyond a petty concern for logic, rationality, coherence, consistency, and principle, and which instead follows after pure unthinking pragmatism.
Denominationalism shares with presbyterianism a concern for the unity of the Body of Christ and that there should be mutual accountability between congregations. Unlike presbyterianism, however, which applies these biblical principles consistently and takes them to the logical conclusion that all de jure churches are required to be united in one worldwide church, denominationalism applies these principles only to an arbitrary point and then stops, switching suddenly after that point to a congregationalist sort of independency. So there is mutual accountability and biblical unity between congregations within a particular denomination, but between the various denominations the biblical rules are broken. And denominations can come in all sizes, so that it can be said that the biblical rules are applied in a completely random and arbitrary manner. So much for consistency and principle!
Similarly, denominationalism shares with congregationalism a concern to preserve complete jurisdictional independence between churches and to not allow one church to lord it over another by sharing authority together with it in a common council, but it only follows through with this principle at the higher levels of church power. Denominations are independent from each other, while individual congregations within each denomination are mutually inter-dependent and lord authority over each other in presbyteries and other assemblies. Thus, the principle of independency is applied in a completely arbitrary and inconsistent manner with no rhyme or reason at all.
In short, denominationalism has the advantage of embracing multiple contradictory principles and then breaking them all. In doing this, it not only defies reason, but it also ensures that it cannot possibly be a biblical position. Congregationalists have developed a particular interpretation of biblical texts to support their point of view, and presbyterians have developed an alternate interpretive scheme, but denominationalism is set up in such a way that it is bound to be unbiblical at some point or another no matter what interpretive scheme is applied. If the Bible teaches the independence and autonomy of congregations, denominationalism both affirms and breaks this principle. If the Bible teaches that there is one Body of Christ that should exist in formal unity and mutual accountability, denominationalism both affirms and breaks this principle as well. Consistency is nowhere in sight at all no matter how one looks at it. Ralph Waldo Emerson would have been proud.
In conclusion, if you want a system of church government that defies both reason and Scripture, embraces and yet satisfies no principles at all, and is completely inconsistent and arbitrary, denominationalism is definitely the choice for you.
For a serious argument in favor of presbyterian church government, see here.