I have been searching on the net for the term dependent and independent samples, but I couldn't find proper definition, nor to conclude what exactly it is. It would be nice if anyone of you could explain it and bestow us with the knowledge.
Reading all the answers and comments, it's clear that we have a bit of a Catch-22 here. People can't answer the question without more context, but the question seems to be asking for that context.
So, I'm going to take a shot at this, trying to guess what Serenity Stack Holder means.
Two samples (or more than two) are dependent if they are somehow connected, not by having similar results necessarily, but by having one result in some way depend on the other result. For example, suppose I am interested in comparing the heights of men and women. If I randomly pick 50 women and 50 men from some population, the samples are independent, because what one person's height is has no bearing on another person's height. One thing gives you no information about the other. However, if I picked 50 heterosexual couples, the two samples would not be independent, because people tend to marry people of similar height.
I hope this helps!
@whuber is right that we need a little bit more context to decipher what you mean by "samples." If you mean "samples" in the sense of "the result of doing sampling," and thus you're using the term as a synonym of "realizations", then the following applies:
Samples are dependent conditional on some (or possibly no) prior knowledge if and only if knowing something about one sample could tell you something new about the other sample.
The most common case is where samples $x_1, x_2$ are assumed to be "independently and identically distributed" according to a distribution $D$. In that case, given that you know $D$ is say a normal distribution with mean zero and variance one, knowing that the value of $x_1$ is $1.2$ your belief about $x_2$ is still that it follows $D$. Without knowing $D$, however, but knowing that the samples were iid from some normal distribution, leaves the samples clearly dependent: knowing something about one tells you something about $D$, which tells you something about the other.
Without assuming dependence or independence between samples, it's impossible to know whether they are dependent or independent, but one can often make good guesses by trying to find patterns. Correlation, the example above, is just one such pattern.
Maybe a list with easy cases is a start:
if your samples are correlated, they are not independent (but you cannot conclude the other way round).
(kind of obvious): if one sample influences the other, they are not independent
if you know a cause that influences both samples, they are not independent
if you are able refine a prediction of whatever you are interested for one sample once you know the result for another sample (i.e. better than guessing from the overall distribution), then samples are not independent.
it is always difficult to argue independence: Imagine you study hair color and to do that you grab people every 30 min from the street in front of your office. You may consider the persons independent: no way to predict another persons hair color better than guessing the average hair colour. But how do you know (prove!) that you did not just miss the right model to predict hair colour?
(In)dependence may be discussed with regard to the scope of the study: now your colleague at the other side of the globe joins your study and sends you hair colors of people in front of his office. Now you can predict upcoming hairs better than general guessing from the last hair color you collected: hair colour is not independent of geographical region.
You may say that the population to be studied needs to be well defined in order to argue (in)dependence: is it "hair colours appearing in front of my office" or is it "hair colour of humans"?