Trials in an experiment are independent if the likelihood of each possible outcome does not change from trial to trial. For example, if you toss a coin fifty times, the coin tosses are independent trials because the outcome of one toss (heads or tails) does not affect the likelihood of getting a heads or tails on the next toss.
However, suppose you draw cards one at a time from a standard deck of cards without putting the cards back into the deck. Your chance of drawing an ace on the first draw is 4/52. If you draw an ace on the first draw, your chance of drawing an ace on the second draw changes from 4/52 to 3/51. Thus the two trials are dependent, not independent.
In the quality setting, if a marketing analyst orally asks a yes/no question to a focus group in a room, each person's answer can be affected by the answers from the other people who have already spoken. Thus the outcomes of the trials (question-answer) are dependent, not independent.
The type of statistical analysis you use to assess data could depend on whether the trials are dependent or independent. For example, independent trials are an important assumption for assessing process capability using a binomial distribution, when each trial has only two possible outcomes.
Suppose an automotive company manufactures precision metal parts for gas turbines. Before shipping, inspectors randomly choose parts and assess their dimensions using a laser gage. Based on the gage results, they pass or fail each part. Because each decision to pass or fail a part is independent, they can do a process capability with a binomial distribution to estimate whether the percentage of defective parts falls within the company specifications.