The word "relation" in mathematics can mean different things.

We can relate mathematical expressions.  For example, equality is an example of relating two expressions, as is inequality.

We can also discuss relations in terms of ordered pairs.  Given two sets, one being the domain and the other being the range, we can relate each member of the domain with one or more elements in the range. 

We can write this second use of the term relations in numerous ways, including as a set of ordered pairs or as by using a rule to describe how to find an element in the range given an element in the domain.

Some examples of relations:

{(1, 4), (2, 3), (1, 3), (4, 6)}

y = x + 1

We must be careful not to confuse the concept of relations with that of functions.  All functions are relations, but not all relations are functions.  In the above examples, both describe relations, but only the second example, y = x + 1 is a function, since the first set of ordered pairs does not define a unique range value for every element in the domain.  See the entry about functions for more information on the difference between relations and functions.