One commonly used (and perhaps commonly experienced) Type I error occurs in the use of home pregnancy tests. When the test returns a positive result (meaning that the woman is indeed pregnant), but in fact no pregnancy exists, the test has returned a false positive, meaning a Type I error has occurred. In this instance the null hypothesis would be no pregnancy -- no change from the status quo. The null hypothesis that there is no pregnancy is falsely rejected when the pregnancy test returns a positive result without a pregnancy truly existing.

When a car is taken to the mechanic for a tune-up and basic diagnostic, and is given a clean bill of health just days before the radiator blows or the transmission falls out, a false negative or Type II error has occurred. In this case, the null hypothesis -- that the car is fine and no serious problems have developed...

It is impossible in this scenario to figure out exactly how the error occurred, but regardless the information that the mechanic was able to obtain and/or upon which his decision was based led to the false belief that the status quo had been preserved and the car was fine.
This also illustrates precisely why it is improper to use the terms "proof" and "truth" when discussing data analysis and research findings. A true fact can only be established if it has been tested in every possible scenario -- a practical impossibility. On the other hand, only one instance in which the hypothesis does not hold true is needed to falsify something, and it is always possible that this instance lurks just around the corner.