In probability theory, a sequence or other collection of random variables is<unk>  and<unk><unk>  (i.i.d.) if each has the same probability distribution as the others and all are mutually<unk> .
The abbreviation i.i.d. is particularly common in statistics (often as iid, sometimes written IID), where observations in a sample are often assumed to be (more-or-less) i.i.d. for the purposes of statistical inference.  The assumption (or requirement) that observations be i.i.d. tends to simplify the underlying mathematics of many statistical methods.  However, in practical applications this may or may not be realistic.
This is important in the classical form of the central limit theorem, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution, becoming acceptably close when sample size n > 29.
Independent and identically-distributed random variables
Random variable
Dependent and independent variables
Dose-response relationship
Biased sample
Degrees of freedom 
Covariance matrix
Bias of an estimator