Confusing language and terminology is a big part of what makes statistics confusing. Each Binomial Trial -- also known as a Bernoulli Trial -- is a random experiment with only 2 possible outcomes, called "success" and "failure". The Probability of success is the same every time the experiment is conducted.
A coin flip illustrates this perfectly. You either get "heads" or "tails", and the Probability of each coin flip (Binomial Trial) is always 50% heads (or 50% tails).
In a Binomial Trial, each trial is counted as either a success or failure. And a success is defined as what we want to count. Let's say we are performing quality control in a manufacturing process. We are counting defects. Every time we find a defect, we add 1 to the count of "successes".
I always found that confusing, so in the book, instead of saying "success" or "failure", I suggest saying "yes" or "no".
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.