Can someone explain what it means for an observation, from a group of data, to be independently and identically distributed? I don't quite understand what the difference is between an observation being independent and it also being  identically distributed. Can someone explain please?
 A: It must be pointed out that observations can be neither "independent" nor "identically distributed". This are characteristics of random variables whose observed realizations we call "observations".  In informal communication though, we do tend to say "independent and/or identically distributed observations", as a shortcut of "records of realizations of independent and identically distributed random variables".
To the issues of the question, both "independence" and "identically distributed" are concepts that characterize a random variable in relation to other random variables. Let $X$ and $Y$ be two random variables with marginal distributions represented by the Cumulative Distribution Functions (CDFs) $F_X(x)$ and $G_Y(y)$. Denote also $F_{X|Y}(x\mid y)$ and $G_{Y|X}(y\mid x)$ the conditional CDFs. Finally, the joint distribution function is $H_{XY}(x, y)$.
IDENTICALLY DISTRIBUTED 
The random variables $X$ and $Y$ are identically distributed iff
$$F_X(x) =G_Y(y),  \forall \{x=y\}$$
i.e. if they have the same marginal probability distribution (not just belonging to the same "family"). This extends immediately to more than two random variables.
INDEPENDENCE
If one digs deep enough, one will find alternative definitions of "stochastic independence" or "independence in probability", but it is my impression that the essence is the same: informally, two random variables are independent if the behavior of the one does not alter the marginal probability distribution of the other. 
The random variables $X$ and $Y$ are independent iff
$$G_{Y|X}(y\mid x) = G_Y(y) \;\; \Big[\Rightarrow F_{X|Y}(x\mid y) = F_X(x)\Big]$$
Essentially the above is a derivative of the more fundamental definition which is that the joint probability distribution equals the product of the marginals:
$$H_{XY}(x, y) = F_X(x)G_Y(y) $$
but the formulation using the conditional distributions reflects the informal description of independence.
When the random variables under examination are more then two, here things become tricky: here at least two different notions of independence arise, "pair-wise independence" and "mutual independence". When we talk about "an i.i.d sample of size $n$" without other qualifications, then the concept of independence invoked is that of mutual independence, which says that the joint distribution function of the $n$ random variables equals the product of the marginal distribution functions.  
In discussing independence, we have not assumed that the random variables are identically distributed (we used different letters for the CDFs to stress this). Having the same marginal probability distribution has nothing to do with whether two random variables are independent or not. They can have the same distribution and be dependent or independent, and they can have different distributions and be dependent or independent.  
I do not argue that the above are the fully rigorous statements in mathematical statistics terms, but I believe they handle the concepts adequately.  
A: *

*Random variables would be identically distributed if they have the same distribution function (if their distribution has the same shape). But that doesn't imply independence. 
So a pair of correlated random variables could have the same marginal distribution function, but are not independent.
The two sets of random variables $X_i$ and $Y_i$ (which x1 and y1 below are samples from) have the same marginal distribution, but are not independent:


*Random variables would be independent if the conditional distribution ("distribution of one when you know the other") would be the same as the marginal distribution ("distribution of one without any information on the other"). That doesn't imply identically distributed.
For example, if I pour some sugar into a cup and roll a die on the table, whether the number of grains of sugar is odd or even and the outcome on the die would presumably be independent.
These two samples below are drawn from different distributions, but are drawn independently. If you drew a new pair, and knew that say the observed value of $W$ was between 0.05 and 0.10, it would tell you nothing further about what the corresponding $Y$ was than you already knew.

