What did Newton and Leibniz actually discover? Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a limit with its $\epsilon$-$\delta$ definition) that seem to have been developed only much later (in this case in the late $18$th and early $19$th century).
Hence the question - what is it that Newton and Leibniz discovered?  
 A: Of course Archimedes had "exhaustion" principles to compute areas and surfaces of curved bodies using limiting arguments, like the so-called Cavalieri principle.
The essential point of Newton's and Leibniz' calculus is that they had the notion of "function". This notion is immediately tied to Descartes' idea of "coordinate system", which freed people of considering things like $x^2$ as pieces of area, and the like. Furthermore it was now allowed to consider time $t$ as a variable entering quantitative discussions.
Calculus was put on the map when Newton and Leibniz discovered that any function $f$ has certain natural concomitant functions $f'$, $f''$, as well as antiderivatives, and that the connection between these can be interpreted in a geometrical or physical way.
A: I'm not a historian nor have studied these things deeply. So, please make corrections in commentaries/edits when necessary.
From what I've read, it was more or less like this:
Calculus was discovered/created independently by Newton and Leibniz.


*

*Leibniz started with sequences of numbers, and after discovering some results about them he sought a tutor in mathematics. He found Huygens and told him about his discoveries and asked to be tutored. Huygens tested him with a problem of finding the sum of a given sequence. Leibniz easily solved the problem proposed. Huygens accepted to tutor him and taught him analytic geometry and presented texts of Pascal and Fermat to him. Then Leibniz discovered/created calculus.

*Newton was tutored by Barrow, who was a very competent classical geometer (i.e. in the style of the Ancient Greeks). Barrow was very interested in chronology and because of that he studied kinematics, optics and mathematics. Barrow discovered the fundamental theorem of calculus, but he did not know much of algebra and analytic geometry. Newton, being very comfortable with algebra and analytic geometry, after having learned the "geometric calculus" of Barrow soon turned it into a machine of solving problems. In this sense, Newton discovered/created calculus. 
Note: It's very interesting that Newton developed many "versions" of calculus: one time he even sought to divorce it from algebra and create a "geometric calculus" more powerful than that of Barrow, by using things like similar triangles and "ultimate ratios", but he also had one more "algebraic" with relied more on polynomials and series, and he experimented many ways of presenting his calculus from "the start".
Much before: Eudoxus created the method of exhaustion which can be used to prove some things that today are proved via the theory of limits. Archimedes discovered infinitesimals and used them to obtain many results in geometry, but his published proofs did not mention infinitesimals, and his letter to a friend, in which he explained infinitesimals, was lost until recently and so infinitesimals had to be rediscovered in Europe (much later!) and were then used to calculate areas and volumes. Descartes and Fermat developed analytic geometry which was maybe crucial for Newton and Leibniz to discover the calculus. Limit theory was developed much later. More recently, infinitesimals were finally accepted after a long history of rejection.
A: To understand the relation between Leibniz's theory, on the one hand, and modern ϵ-δ definitions and constructions of the continuum, on the other, it is helpful to make the following distinction. The distinction is between the following two items.
(1) On the one hand, we have the procedures employed by Leibniz and other pioneers of the calculus, and their procedural moves in deriving various new results such as Leibniz's derivation of the product formula for differentiation, and Euler's various solutions of the Basel problem, to give some elementary examples. These procedures can be characterized as syntactic material because at this stage in the development of mathematics, its practitioners were not overly concerned with semantic issues (see below).
(2) On the other hand, we have the issue of the ontology of mathematical objects, i.e., what are the basic objects mathematics uses, how one axiomatizes their behavior and justifies their "existence" relative to appropriate foundational theories. These semantic issues only began to emerge starting around 1870, and resulted in the modern foundations for analysis in particular and much of modern mathematics in general.
To answer your question, I would say that the contributions of Leibniz and others would today be viewed as centering on the syntactic side of the mathematical endeavor.  They developed the calculus in the sense that they understood its procedures and syntax, which remain today largely as they have been developed by Leibniz, Euler, Lagrange, and others.
The semantic side of the story did not emerge until much later.  This observation is sometimes expressed by claiming that the pioneers of the calculus were not "rigorous" but this I believe is a misleading statement: Gauss and Dirichlet similarly did not have access to modern sematic theories and in this sense would also not be "rigorous", but in fact there are almost no errors in their work. For this reason it is more meaningful to acknowledge the syntactic contribution of the pioneers of the calculus than to apply the essentially anachronistic straitjacket of "mathematical rigor" in analyzing their work.
A: This is actually quite a complicated question, since it spans two whole careers.
Some say calculus was not discovered by Newton and Leibniz because Archimedes and others did it first.  That's a somewhat simple-minded view.  Archimedes solved a whole slew of problems that would now be done by integral calculus, and his methods had things in common with what's now taught in calculus ("now" = since about 300 years ago), but his concepts were in a number of ways different, and I don't think he had anything like the "fundamental theorem".
I'm fairly sure Leibniz introduced the "Leibniz" notation, in which $dy$ and $dx$ are corresponding infinitely small increments of $y$ and $x$, and the integral notation $\int f(x)\,dx$.  I suspect Newton and Leibniz were the first to systematically exploit the fundamental theorem.  And the word "systematic" is also important here: Newton and Leibniz made the computation of derivatives and integrals systematic.
A: I would like to suggest a book written by an italian mathematician (Lucio Russo), namely "The forgotten revolution: how science was born in 300 BC and why it had to be reborn". It contains a lot of interesting things some of which are connected to the story of infinitesimal calculus spreading a (different) light on the works by Newton, Leibniz and Co.
A: from my understanding, Newton's binomial said that the delta x should always have lim -> 0 because it's in the denominator, but once you got rid of it down there, delta x=0 is possible (and then you got rid of all the delta x in the numerator), thus, making the computation of derivatives systematic and easy.
