A trial in an experiment is independent if the likelihood of each possible outcome does not change from trial to trial. For example, if you toss a coin fifty times, each coin toss is an independent trial, 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.
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 a 1 proportion test, when each trial has only two possible outcomes.