A data set is not just a list of numbers or values; there is some context associated with it, usually the units, or what type of measurement is used, or perhaps some kind of descriptor.
A variable is any characteristic of the individual members of the population that can be measured. A variable of interest can take on different values for each member of the population.
EXAMPLE
For example, suppose we are interested in the variable of height for a group of people. This could vary from person to person because people have different heights.A distribution is a way to visually show how many times a variable takes a certain value; it is the values the variable takes and how often they show up. There are many kinds of distributions:
Types of Distributions  Description  Examples  

Frequency tables  Can visually show how often a variable takes on a certain value  
Qualitative Data  The variables in these distributions are categories. 
Bar Graphs Pie Charts Dot Plots 

Quantitative Data  The variables in these distributions are measures of values or counts. 
StemandLeaf Plots Dot Plots Histograms Line Charts TimeSeries Diagrams 

Mathematical Rules  Can visually show variables through a certain pattern and are not strictly datadriven. 
Normal Distribution Poisson Distribution 
Why are there so many different kinds of distributions? The point of a distribution is to make the datapossibly a large data set that is unwieldysimpler to understand. You want to make it easy for yourself and your readers to understand. Therefore, different kinds of distributions will lend themselves better to different types of data sets.
EXAMPLE
A dot plot is better for data that are close together and doesn't have a lot of values, whereas certain other distributions are better for larger data sets. A histogram is better than a dot plot when the data is very spread out.You can determine which kind of distribution to use based on the kind of data you have.
Source: Adapted from Sophia tutorial by Jonathan Osters.