Independent observations
For most tests, it is assumed that the observations are independent. That is the results for one subject* are not affected by another. Examples of data which is not independent are repeated measures on the same subject (use the specific tests for this type of experiment) and observations over time (check the Durbin Watson test for regression). Another situation where observations are not independent is when subjects are nested within groups with a common influence e.g. children within classes who may be influenced by the teacher (use multilevel modelling to include class as an extra RANDOM factor). Time series analysis (which allows for non-independent measurements over time) and multilevel modelling are beyond the scope of most students.
*The subject is the unit of interest which could be a person, an observation, a day etc.