As the survival tag is used I'll add an answer offering some examples with a survival analysis flavour.
By a data point, we just mean some observation, i.e. the outcome of one or more variables. For instance, we might have the following in a data set: person 1 in our study is a male and dies at age 58. We could consider that a data point. But in your example it's clear that the data point only consists of the outcome of one variable, e.g. 58.
If we're modelling the time to failure there's an obvious reason for censoring, namely, that we don't necessarily have time to wait for all subjects to fail. Say we're testing the effect of children's vaccines. If we were to conduct a randomized trial, the last of our subjects would die a hundred years or more from now. This naturally introduces censoring, in this case right-censoring, as we would at some point have to say "we don't know how much longer this person will live, we only know that she's still alive". Right-censoring might also occur if people in the randomized trial are lost to follow-up, e.g. they might want to discontinue their participation in the study or move away. These are examples of right-censoring, basically we're interested in the longevity of our subjects but due to practical circumstances we only have censored observations, meaning that for some subjects we will never know when they die, only that at some point in time (the censoring time) they were still alive. Thus, we know that for a censored individual, the data point (time of death) is larger than a certain value (the censoring time).
As an example of left-censoring, consider the following. Let's say that some baboon troop always sleeps in the trees. We want to estimate at what time in the morning they descend from the trees, and let's assume that they do descend every day. We follow them for a number days, however, we like sleeping in, meaning that some days they descend before we even arrive on the scene. If we arrive at 9 a.m. on day $x$ and the baboons already descended, we have left-censored data. We want to know when they descended, but all we have is an upper limit (9 a.m.), because we know that at our time of arrival they had already descended. Analogously, we now know that the data point (time of descent on day $x$) is smaller than a certain value (9. a.m.).
This example is taken from
Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (1993), Statistical Models based on Counting Processes, Springer Series in Statistics, Springer-Verlag, New York.
This book provides a mathematical definition of censoring and is probably not the first book on survival analysis one should get. However, it also has some intuitive examples, as the above.