Main Difference
The main difference between Algebra and Calculus is that the Algebra is a part of mathematics in which letters and other symbols are used to represent numbers and quantities in formulae and equation and Calculus is a branch of mathematics

Algebra
Algebra (from Arabic: الجبر, transliterated “aljabr”, literally meaning “reunion of broken parts”) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in
x
+
2
=
5
{displaystyle x+2=5}
the letter
x
{displaystyle x}
is unknown, but applying additive inverses can reveal its value:
x
=
3
{displaystyle x=3}
. In E = mc2, the letters
E
{displaystyle E}
and
m
{displaystyle m}
are variables, and the letter
c
{displaystyle c}
is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an “algebra”, and the word is used, for example, in the phrases linear algebra and algebraic topology.
A mathematician who does research in algebra is called an algebraist.

Calculus
Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals”, is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a welldefined limit.Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally “small pebble” (this meaning is kept in medicine). Because such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.

Algebra (noun)
A system for computation using letters or other symbols to represent numbers, with rules for manipulating these symbols.

Algebra (noun)
The surgical treatment of a dislocated or fractured bone. Also : a dislocation or fracture.

Algebra (noun)
The study of algebraic structures.

Algebra (noun)
A universal algebra.

Algebra (noun)
An algebraic structure consisting of a module over a commutative ring (or a vector space over a field) along with an additional binary operation that is bilinear over module (or vector) addition and scalar multiplication.
“algebra over a fieldalgebra over a ring”

Algebra (noun)
A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and complements (and thereby also under intersections and differences).
“field of sets”

Algebra (noun)
One of several other types of mathematical structure.

Algebra (noun)
A system or process, that is like algebra by substituting one thing for another, or in using signs, symbols, etc., to represent concepts or ideas.

Calculus (noun)
Calculation; computation.

Calculus (noun)
Any formal system in which symbolic expressions are manipulated according to fixed rules.
“lambda calculus”
“predicate calculus”

Calculus (noun)
Differential calculus and integral calculus considered as a single subject; analysis.

Calculus (noun)
A stony concretion that forms in a bodily organ.
“renal calculus ( = kidney stone)”

Calculus (noun)
Deposits of calcium phosphate salts on teeth.

Calculus (noun)
A decisionmaking method, especially one appropriate for a specialised realm.