Elaboration of Arthur Cayley's definition of group? In 1854, Arthur Cayley, the British Mathematician, gave the modern definition of group
for the first time:

“A set of symbols all of them different, and such that the product of any two of
  them (no matter in what order), or the product of any one of them into itself,
  belongs to the set, is said to be a group. These symbols are not in general
  convertible [commutative], but are associative.”

What is the exact elaboration/illustration of this definition?
A few questions arises from definition these are:
1.(no matter in what order) but symbols are not in general
convertible [commutative].
2.What is the meaning of product here?result of multiplication?
 A: This is from an 1854 paper by Cayley, "On the Theory of Groups as depending on the Symbolical Equation $\theta^n=1,\!"$ published in the The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Series 4, vol. 7, 1854, issue 42, pp. 40-47. (I think there were additional parts published later.)  You can read the article at: https://books.google.com/books?id=dt8ClkIoBIQC&pg=PA40 -- it's quite remarkable.
Here's an excerpt:

A: The formal definition:
A group is a non-empty set $G$ together with a binary operation $*$ such that
I. There exists an identity element $e \in G$ such that $a*e = e*a = a$ for every $a \in G$.
II. For each $a \in G$, there exists an inverse element $a^{-1} \in G$ such that $a*a^{-1} = a^{-1}*a = e$.
III. $*$ is associative, i.e. $(a*b)*c = a*(b*c)$ for all $a,b,c \in G$.
We often use multiplicative notation $ab = a * b$, as the passage you quoted suggests. This should clear up your second question. The "result of multiplication" is just whatever you get when you apply the binary operation $*$ to $a$ and $b$. Your first question means that for some groups, it is not true that $ab = ba$ for all elements $a,b$ in the group. In other words the order in which you multiply the elements matters.
A prototypical example of a group is $\mathbb{Z}$ under addition. $0$ is the identity element, $-n$ is the inverse of $n$, and and addition is associative. This group is also commutative, meaning $a + b = b + a$ for all $a,b \in \mathbb{Z}$. An example of a noncommutative group is the group of invertible functions from $\mathbb{R}$ to $\mathbb{R}$. This is a group under function composition. Function composition is associative, $f(x) = x$ is the identity, and by definition functions in this group have an inverse. However, function composition is not commutative. For example, if $f(x) = x + 1$ and $g(x) = x^2$, then
$$f(g(x)) = x^2 + 1 \neq (x + 1)^2 = g(f(x)).$$
Hope this clears up any confusion.
A: It may help to put it in the context of the 19th Century English school of "symbolic algebra".
See George Peacock and his A Treatise on Algebra (1830) :

[page vi] Preface. Algebra has always been considered as merely such a
  modification of Arithmetic as arose from the use of symbolical language [...].[page xi] the fundamental operations of Algebra are altogether symbolical, and we might proceed to deduce symbolical results and equivalent forms by means of them without any regard to the principles of any other science [...].
[page 1] 1. Algebra may be defined to be, the science of general reasoning by symbolical language.


Cayley's definition is "abstract", in the sense that it is not intended to have an arithmetical interpretation:

A symbol $\theta \phi$ denotes the compound operation, the performance of which is equivalent to the performance, first of the operation $\phi$, and then of the operation $\theta$; $\theta \phi$ is of course in general different from $\phi \theta$ [emphasis added].

Thus :
1) the "product" is the composition of the "basic" objects : $\theta, \phi$ called symbols of operation;
2) the composition is explicitly not intended to be commutative.
