The censoring mechanism can be thought of as the reason why a subject's (could be a person in a medical trial, a component in an electrical/mechanical apparatus, etc.) time to a particular event occurring is not completely known.
So for example in medical trials testing whether a particular drug can extend people's lives, the censoring mechanisms could be:
People leaving the study (you only know the age at which a person left the trial, not the age of death)
The end of the trial (decided by the investigators; or funding ends) while some individuals are still alive
In both the examples above, you do not get to measure the complete future lifetime of all the individuals in the trial.
In general, the difference between truncation and censoring is:
Truncated values are those that are not included/reported should they exceed/be lower than some limit.
Censored values are those that are reported to be above or below a certain value.
So the practical difference between truncation and censoring is that under truncation, the values below/above a certain value are not included in the data set, whereas under censoring they are included as either an upper and/or lower bound.
Using the ideas above, suppose there is a medical device that measures the concentration of some toxin in people’s blood. Suppose that the sensitivity of the device is such that it can only detect the toxin if the concentration is above 0.1g/ml.
Suppose that the investigators know that all people that they test do have this toxin in the blood.
In this case, if the device indicates that a particular person does not have the toxin in their blood, it means that the concentration is less than 0.1g/ml. In this case the observation is Left Censored as we do not know the precise concentration, but we do know that the concentration is less than 0.1g/ml.
Suppose that the investigators test people at random, i.e. they do not know whether the person has the toxin in their blood, and that they are only interested in people who are reported to have the toxin in the blood as defined by the device.
In this case if the device indicates that the person does not have the toxin, i.e. the concentration reported is below 0.1g/ml, the investigators will not include this person in their report and they will be Left Truncated.