Imagine that a system want to detect apples and oranges in a basket of fruits. System can pick a fruit, extract some property of it (e.g weight of that fruit).
Suppose System has a Teacher! that teaches the system which objects are apples and which are oranges. This is an example of a supervised classification problem. It is supervised because we have labeled examples. It is classification because the output is a prediction of which class our object belongs too.
In this example we consider 3 features (properties/explanatory variables):
(0 mean No, and 1 means Yes)
So to represent an apple/orange we have a series(called vector) of 3 properties(often called a feature vector)
(e.g [0,0,1] mean that this fruit weight is Not greater that .5gram, and it's size is Not greater than 10cm and the color of it is red)
So, We pick 10 fruit randomly and measure their properties. The teacher (human) then labels each fruit manually as apple =>  or orange => .
eg) Teacher select a fruit that is apple. The representation of this apple for system could be something like this: [1, 1, 1] => , This means that, this fruit has 1.weight greater that .5gram, 2.size greater than 10cm and 3. the color of this fruit is red and finally it is an apple(=> )
So for all of 10 fruits, the teacher label each fruit as apple[=>1] or orange[=>2] and the system found their properties. as you guess we have a series of vector(that called it matrix) to represent whole 10 fruits.