We keep that we have 821 of something minus 5*25 of that thing (which we recall is called "cents").
This is all that is necessary to answer the question, the rest can and should be thrown away. Then the problem
25 people each drink a liter of water a day. If they began with 821 liters, how many liters are left after 5 days?
is the same problem, to the mathematician, who doesn't care how a liter of water and a penny are different.
Now, you've probably known all this for many years. We learn to throw away information before we even know what we're doing. Let's get to more exciting examples.
In problems of counting and arithmetic all that matters is number, the objects can be anything. In these cases mathematicians can represent a collection of objects by a collection of points, a point being a sort of universal object for counting. In mathematics, collections of objects are called sets, and frequently we regard these as collections of points, since that is enough information to allow counting.
When we begin with a set of objects and consider each object to be a point, we are parameterizing the set we began with.
Information that tells how mathematical objects are related is called structure. Sometimes we are interested in more structure than just number. For example, suppose we want to parameterize family members while retaining the familial relationships between members (such as parent, sibling, etc.). We might begin with a set of points, one for each family member. What sort of structure do we need to add to identify familial relationships?
