BigO notations (in Computer Science theory) allows to categorize for a given set of inputs, what order of magnitude the running time (and/or space time) an algorithm would take. It allows in making the best decisions to choose given algorithm for a particular problem or know it limitations in advance.
Consider the trivial examples of various sorting and searching algorithms, The BigO notation would allow to describe why merge sort may be better or worse than quick sort and/or better or worse than bubble sort.
As a user of searching algorithms, for instance, I would ideally want to use a data structure which runs in O(1) (for example hash table or lookup table). Or is it acceptable for running time to be O(log n), which is applicable on a sorted binary tree search. Or for sorting examples, will there be memory constraints or running time constraints or both. If my input size is smaller and my memory requirements is smaller, would it will be acceptable to choose a slow running sort?
As part of algorithm analysis and design in Computer Science, we have to understand what is the complexity of each algorithm in terms of Big O notation (or how to come up with the Big O notation in first place for a given algo). Then you make trade offs by analyzing various algorithms for a particular solution of a problem.
