Geometric(p) models the total number of trials that will occur before the first success in a set of binomial trials, given that p is the probability of a trial succeeding. The total number of trials is equal to the total number of failures plus 1 (the first success). Examples of the Geometric distribution are shown below:
I select a card from a pack (no jokers) and guess its suit before looking at it. The total number of guesses I will have to get it right can be estimated as Geometric(25%). The Geometric distribution assumes that p is constant with each trial i.e. that I cannot get any better at guessing with each failure, nor does my problem change (so I'll have to put the card back and reshuffle). It also assumes that I will doggedly carry on, even if it takes me a hundred failures before I succeed. Thus, some caution is needed in its application.
Dry oil wells
The Geometric distribution is sometimes quoted as useful to estimate the total number of wells an oil company will drill in a particular section before getting a producing well. The total number of wells is equal to all dry wells plus the 1 producing well. That would, however, be assuming that a) the company does not learn from its mistakes; and b) it has the money and obstinacy to keep drilling new wells despite the cost.
More sensible example
You need to purchase some item, conduct a test or operation on that item, and if you fail, go and buy another. For example, you need to find a cow with disease X, but a definitive test involves an expensive procedure, so you randomly select a cow, test it, etc. The number of cows you'll need to buy is equal to Geometric(p) where p is the prevalence of disease X among the cows.
Note that this would not work if you bought cows in batches. For example, if you bought batches of 5 cows. Then the number you'd have to buy to get an infected cow is: =(Geometric(P))*5, where P = 1-(1-p)5, i.e. the probability that a batch of 5 cows contains at least one infected.
The Geometric distribution is a special case of the Negative Binomial for s = 1 i.e. Geometric(p) = NegBinomial(p, 1), which means that the sum of s independent Geometric(p) distributions = NegBinomial(p,s). The Geometric distribution is the discrete analogue of the Exponential distribution, and gets its name because its probability mass function is a geometric progression. The Geometric distribution is occasionally called a Furry distribution.
An alternative formulation of the Geometric distribution models the number of failures to observe the first success (instead of total number trials, which is the number of failures plus 1). In this formulation, the probability mass function is given by