The Confederate States's Constitution (and therefore the structure of its government) was almost identical to that of the United States - with good reason, as the Confederates at least initially saw themselves as "conservative" revolutionaries, revolting to maintain the status quo antebellum rather than re-make southern society. The Confederate executive had the same cabinet positions - War, State, Treasury, etc. - as its Northern counterpart. The bicameral structure of the Congress - Senate and House of Representatives - was preserved. Though the war prevented a Confederate Supreme Court from ever sitting, it was officially established by Article III.
The Wikipedia article highlights the differences between the two Constitutions. Some are cosmetic, others are telling. Here are a few of the latter type:
The Confederate constitution outlawed the international slave trade outright, rather than simply specifying a point at which the Congress would be allowed to do so. As was pointed out long before the war, there were commercial reasons for ardently pro-slavery politicians to support the prohibition of the international trade - namely, that the "natural increase" of the slave population was sufficient to ensure an adequate supply of chattel labor, that the international trade had become hopelessly unpopular, and that prohibiting the trade made the slaves currently in the States even more valuable by restricting supply.
The Confederate President had a line-item veto. This was an outgrowth of Southern complaints about excessive Federal spending. The line item veto allowed the President effectively to remove portions of appropriations bills that he disliked.
The Confederate Constitution denied the Congress the power to legislate against slavery. Article I, Section 9 (emphasis added): "No bill of attainder, ex post facto law, or law denying or impairing the right of property in negro slaves shall be passed."
The Confederate President was elected for a six-year term, and could only serve once. For the first year after the Constitution was signed, Jefferson Davis served as "provisional" President of the Confederacy. He ran unopposed in the Presidential Election of November 1861, and was inaugurated as President proper in February of 1862. Congress was similar: For the first year, it consisted of "provisional" Representatives and Senators. Subsequently, two Congresses sat after elections in Autumn 1861 and 1863-64. The Confederate government dissolved before the Second Confederate Congress could complete its two-year term.