Calculus and algebra are two fields of study in mathematics.
Calculus deals with the study of change in relations a.k.a equations. Calculus is split in to two parts, one studies the rates of change and slopes of curves and the other studies the area underneath curves and the accumulation of quantities. Hence differential and integral calculus.
Algebra on the other hand is the study of operations and what you can do with them to solve equations a.k.a "dubious relations".
The word calculus comes from the latin word "calculus" which means little stone. Outside the english speaking world, in the majority of countries, calculus is called analysis. Calculus was invented (or discovered) at the same time and independently by Isaac Newton and Gottfried Wilhelm Leibniz. Leibniz had a very good opinion about Newton and wanted to collaborate but Newton got all "super star" and obnoxious. No wonder Newton died a virgin. Leibniz's mathematical notation for calculus is used today.
The word algebra comes from arabic. Algebra is the latin version of the arab word Al-Jabr. This came from the title of a book, Hidab al-jabr wal-muqubala, written in Iraq roughly in 840 A.D. by the arab math man Mohammed ibn-Musa al-Khowarizmi. Al-jabr in the book named the basic operation of subtracting terms from one side of an equation to the other. For example: x - 30 = 12 becomes x = 42.
For a more complete history of mathematics starting from ancient time go here: