I should say at the outset, you can probably find someone from any stripe of Christianity who would call herself a Presbyterian. There are both theologically conservative Presbyterian denominations, e.g. the Orthodox Presbyterian Church (OPC) or the Presbyterian Church in America (PCA), and theologically liberal Presbyterian denominations, e.g. the Presbyterian Church, U.S.A. (PC-USA). I am a member of the PCA and can therefore only give an essentially conservative perspective.
The word “presbyterian” comes from the Greek πρεσβύτερος (presbyteros) and describes the church polity (or church government) of all Presbyterian denominations, which differs from the episcopalian or congregational forms. Where episcopalian polity holds to a “top down” hierarchy of bishops and congregational polity affirms the “bottom up” independence of individual churches, presbyterian polity attempts to find a middle ground with a plurality of elders (a “session”) leading each congregation and groups of elders from several congregations forming wider “presbyteries” to keep the churches mutually accountable to one another.
However, though unique, this form of church government could hardly be described as the main thing that makes Presbyterians different from other forms of Christianity. Presbyterian theology, including its church polity, follows the ideas of John Calvin and can therefore be called “Calvinist” or “Reformed” theology. (As such, Presbyterian theology is held in common with other Reformed denominations, some of which do not hold to the presbyterian form of church government. Therefore, though I’ll continue to use the word “Presbyterian” below, this should not be taken to imply that any of the following is unique to Presbyterianism.)
One of the chief tenets of Presbyterianism is called “covenant theology”. Covenant theology posits that a single Covenant of Grace runs through both the Old and New Testaments and that the Christian church is therefore a continuation of national Israel, having been grafted as wild olive branches into the same olive tree (Romans 11:11–24). This view is held in contrast to the dispensationalist view held by some other Christian denominations, which affirms that God’s promises and plans were for national Israel in the Old Testament and the Christian church in the New Testament.
Largely because of covenant theology, Presbyterianism places a premium on expository preaching through the entire Bible, finding Christ and the Christian gospel just as clearly in the Old Testament as in the New.
Also arising largely out of covenant theology is a unique form of soteriology (i.e. understanding of salvation) including, perhaps most controversially, the concept of God sovereignly predestining those he would save (i.e. his “elect”) from before the creation of the world and then irresistibly drawing those people to himself. This form of soteriology commonly goes under the name “Doctrines of Grace” or under the acronym “TULIP”. It is also sometimes simply called “Calvinism”, though Calvinism should rightly be understood to encompass more than just this concept of Christian salvation.
I’m trying to keep my answer short, but I would be remiss if I did not also mention briefly that Presbyterians affirm infant baptism and tend to be postmillennial or amillennial in their eschatology (understanding of end time prophecy). These ideas, too, are derived from covenant theology (sensing a pattern, yet?).
Conservative Presbyterian denominations all affirm a historic statement of faith called the Westminster Confession of Faith, an excellent and not overly long read if you want to learn more. And if you’d really like to study Presbyterianism, I recommend On Being Presbyterian: Our Beliefs, Practices, And Stories by Sean Michael Lucas. 
This post was originally published on Quora, here: How does Presbyterian theology differ from other forms of Christian theology?