Can someone help to explain the difference between independent and random? In statistics, does independent and random describe the same characteristics? What's the difference between them? We often come across the description like "two independent random variables" or "random sampling". I am wondering what's the exact difference between them. Can someone explain this and give some examples? for instance non-independent but random process?
 A: Random relates to random variable, and independent relates to probabilistic independence. By independence we mean that observing one variable does not tell us anything about the another, or in more formal terms, if $X$ and $Y$ are two random variables, then we say that they are independent if
$$ p_{X,Y}(x, y) = p_X(x)\,p_Y(y) $$
moreover
$$ E(XY) = E(X)E(Y) $$
and their covariance is zero. Random variable $Y$ is dependent on $X$ if it can be written as a function of $X$
$$ Y = f(X) $$
So in this case $Y$ is random and dependent on $X$.
Calling process "non-independent" is pretty misleading - independent of what? I guess you meant that there are some $X_1,\dots,X_k$ independent and identically distributed random variables (check here, or here) that come from some process. By independent we would mean in here that they are independent of each other. There are processes producing dependent random variables, e.g.
$$ X_i = X_{i-1} + \varepsilon $$
where $\varepsilon$ is some random noise. Obviously in such case $X_i$ is dependent on $X_{i-1}$, but it is also random.
A: I'll try to explain it in non-technical terms: A random variable describes an outcome of an experiment; you can not know in advance what the exact outcome will be but you have some information: you know which outcomes are possible and you know, for each outcome, its probability.
For example, if you toss a fair coin then you do not know in advance whether you will get head or tail, but you know that these are the possible outcomes and you know that each has 50% chance of occurrence. 
To explain independence you have to toss two fair coins. After tossing the first coin you know that for the second toss the probabilities of head is still 50% and for tail also. If the first toss has no influence on the probabilities of the second one then both tosses are independent. If the first toss has an influence on the probabilities of the second toss then they are dependent.
An example of dependent tosses is when you glue the two coins together.
A: The notion of independence is relative, while you can be random  by yourself. In your example, you have "two independent random variables", and do not need to talk about several "random sampling".
Suppose you cast a perfect die several times. The outcome $6,5,3,5, 4\ldots$ is a priori random. Knowing the past, you cannot predict the number following 4. Suppose I generate a sequence from the other side of the die: $6\to1$, $3\to4$. I get $1,2,4,2, 3\ldots$. It is as random as the first one. You cannot guess what comes after $3$. But the two sequences are completely dependent. 
If one casts two dice in parallel (without interactions between they), their respective sequences will be random and  independent.
A: Variables are used in all fields of mathematics.  The definitions for independence and randomness of a variable are applied unilaterally to all forms of mathematics, not just to statistics.
For example, the X and Y axes in 2-dimensional Euclidean geometry represent independent variables, however, their values are not (usually) assigned at random.
Two given variables can be random, or independent (of one another), or both, or neither.  Statistics tends to focus on the randomness (more correctly, on probability), and whether or not two variables are independent can have many implications for the probabilities of given outcomes being observed.
You tend to see these two properties (independence and randomness) described together when studying statistics, because both are important to know, and can influence the answer to the question at hand.  However, these properties are not synonymous, and in other fields of mathematics they do not necessarily occur together.
A: When you have a pair of values when the first is randomly generated and the second has any dependence on the first one. e.g. height and weight of a man. There is correlation between them. But they are both random.
A: David Bohm in his work Causality and Chance in Modern Physics (London: Routledge, 1957/1984) describes causality, chance, randomness, and independence:
"In nature nothing remains constant. Everything is in a perpetual state of transformation, motion, and change. However, we discover that nothing simply surges up out of nothing without having antecedents that existed before. Likewise, nothing ever disappears without a trace, in the sense that it gives rise to absolutley nothing existing at later times. .... everything comes from other things and gives rise to other things. This principle is not yet a statement of the existence of causality in nature. To come to causality, the next step is then to note that as we study processes taking place under a wide range of conditions, we discover that inside all of the complexity of change and transformation there are relationships that remain effectively constant. .... At this point, however, we meet a new problem. For the necessity of a causal law is never absolute. Thus, we see that one must conceive of the law of nature as necessary only if one abstracts from contingencies, representing essentially independent factors which may exist outside the scope of things that can be treated by the laws under consideration, and which do not follow necessarily from anything that may be specified under the context of these laws. Such contingencies lead to chance." (pp.1-2)
"The tendency for contingencies lying outside a given context to fluctuate independently of happenings inside that context has demonstrated itself to be so widespread that one may enunciate it as a principle; namely, the principle of randomness. By randomness we mean just that this independence leads to fluctuation of these contingencies in a very complicated way over a wide range of possibliities, but in such a manner that statistical averages have a regular and approximately predictable behaviour."
(p.22)
A: The coin example is a great illustration of a random and independent variable, a good good way to think of a random but dependent variable would be the next card drawn from a seven deck shoe of playing cards, the -likelihood- of any specific numerical outcome changes depending on the cards previously dealt, but until only one value of card remains in the shoe, the value of the card to come next will remain random. 
