Continuous distributions define probabilities associated with continuous variables and discrete distributions define probabilities associated with discrete variables. If a variable can take on any value between 2 specified values it is continuous otherwise it is discrete.
A discrete variable would be what you mentioned in your question - will it rain tomorrow , or if we flip a coin and count the number of heads that is a discrete variable.
An example of a continuous variable would be the weights of pupils in a school. They could be any weight.
