What are statistics, under what circumstances do they go out of date, how do you update them?

Statistics determine the selectivity of the indexes. If an
indexed column has unique values then the selectivity of
that index is more, as opposed to an index with non-unique
values. Query optimizer uses these indexes in determining
whether to choose an index or not while executing a query.

Some situations under which you should update statistics:
1) If there is significant change in the key values in the
index
2) If a large amount of data in an indexed column has been
added, changed, or removed (that is, if the distribution of

key values has changed), or the table has been truncated
using the TRUNCATE TABLE statement and then repopulated
3) Database is upgraded from a previous version