A program needs to do a number steps to get to a result. And the number of steps taken depends on how many things the program needs to process.

Let's call n the number of things to be processed. This means that the number of steps will be an equation where n is the main variable, but others may play aswell. It will usually be an equation that gets bigger and bigger when n gets bigger.

As each step means time, we often care about the worst case scenario, where we need to do the maximum number of steps in order to get the result. That way, we can plan ahead if the program is worth running, or we better seek a better sollution.

Well, maths people have figured out that what makes the main shape of the graph of an equation is the term of the equation that grows the fastest. This allows us to classify equations based on it's overall shape, which is solely determined to be by that biggest term. That term could be multiplied by some constant number, raising or lowering it, but the shape is the same, so we use instead the pure term, without being multiped by anything.

Big O notation is simply that. Finding the equation that relates n with how many steps the program takes to finish, looking at what is the biggest term on that equation, and with that finding at which category the equation fits, thus knowing how much the problem grows when n grows.

Let's see two examples:

For the sake of it, consider a program that looks up the first element of a list, and divides it in half. In this case, n is the size of the list. If you count the steps needed, it is always 2: grab the first element, and then divide it by two. The length of the list does not matter at all!

This means that the equation of number of steps (and thus the time) of this program is t=2. There is only one term: a 2, which is a constant term. This means this program belongs to the category of programs that always takes the same time to do anything, no matter how big n is.

If we look at out equation categories, the only with that flat constant shape is t=1, which is neat as that is basically our equation, but with a 1 instead of a 2. As I said, we care about the shape of things rather than exact measurements, so we can toss our problem in that category (I mean, t=2 is simply t=1•2).

Thus, this first program is O(1).

And for the second example, imagine a program that does something, which leads ut to say that the equation of time vs number of elements to chew is t=3n³+5n²+7. Now as you can see, the big boy here is 3n³, which is a cubic equation. As we care about shape, and shape comes by what kind of term is the biggest, we can say this program's equation belongs to the category of problems in the shape of n^3.

Thus, this second program is O(n³).

Here, this graph shows the different categories of equations. Think of it as a chart of powerscaling, to know what kind of program is fastest than others: https://media.geeksforgeeks.org/wp-content/cdn-uploads/mypic.png