While for a discrete probability distribution one could say that an event with probability zero is impossible (e.g. throwing the dice and getting e.g. 3.5: this has probability zero, and (or because) it is impossible), this cannot be said in the case of a continuous random variable, because then no value would be possible (e.g. to measure the width of an oak leaf, and get e.g. 3.5 cm: this is possible, but the 'exact' value of 3.5cm has probability zero, because there are infinitely many exact values even between 3cm and 4cm. Every single one of these exact values has probability zero. Only an interval, eg. the one between 3cm and 4cm may have a probability greater than zero.). This paradox is resolved by realizing that the probability that X attains some value within an uncountable set (for example an interval) cannot be found by adding the probabilities for individual values. Intuitively you could say, every single exact value has an infinitesimally small probability, but strictly spoken that's what we call zero.
Under an alternative and stronger definition, the term "continuous probability distribution" is reserved for distributions that have probability density functions. These are most precisely called absolutely continuous random variables (see Radon–Nikodym theorem). For a random variable X, being absolutely continuous is equivalent to saying that the probability that X attains a value in any given subset S of its range with Lebesgue measure zero is equal to zero. This does not follow from the condition Pr[X = a] = 0 for all real numbers a, since there are uncountable sets with Lebesgue-measure zero (e.g. the Cantor set).
A random variable with the Cantor distribution is continuous according to the first convention, but according to the second, it is not (absolutely) continuous. Also, it is not discrete nor a weighted average of discrete and absolutely continuous random variables.
In practical applications, random variables are often either discrete or absolutely continuous, although mixtures of the two also arise naturally.
The normal distribution, continuous uniform distribution, Beta distribution, and Gamma distribution are well known absolutely continuous distributions. The normal distribution, also called the Gaussian or the bell curve, is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent variables is approximately normal.