In addition to simply finding the rows to be returned by a query, an index may be able to deliver them in a specific sorted order. This allows a query's ORDER BY specification to be honored without a separate sorting step. Of the index types currently supported by PostgreSQL, only B-tree can produce sorted output - the other index types return matching rows in an unspecified, implementation-dependent order.
The planner will consider satisfying an ORDER BY specification either by scanning an available index that matches the specification, or by scanning the table in physical order and doing an explicit sort. For a query that requires scanning a large fraction of the table, an explicit sort is likely to be faster than using an index because it requires less disk I/O due to following a sequential access pattern. Indexes are more useful when only a few rows need be fetched. An important special case is ORDER BY in combination with LIMIT n: an explicit sort will have to process all the data to identify the first n rows, but if there is an index matching the ORDER BY, the first n rows can be retrieved directly, without scanning the remainder at all.
By default, B-tree indexes store their entries in ascending order with nulls last. This means that a forward scan of an index on column x produces output satisfying ORDER BY x (or more verbosely, ORDER BY x ASC NULLS LAST). The index can also be scanned backward, producing output satisfying ORDER BY x DESC (or more verbosely, ORDER BY x DESC NULLS FIRST, since NULLS FIRST is the default for ORDER BY DESC).
You can adjust the ordering of a B-tree index by including the options ASC, DESC, NULLS FIRST, and/or NULLS LAST when creating the index; for example:
CREATE INDEX test2_info_nulls_low ON test2 (info NULLS FIRST); CREATE INDEX test3_desc_index ON test3 (id DESC NULLS LAST);
An index stored in ascending order with nulls first can satisfy either ORDER BY x ASC NULLS FIRST or ORDER BY x DESC NULLS LAST depending on which direction it is scanned in.
You might wonder why bother providing all four options, when two options together with the possibility of backward scan would cover all the variants of ORDER BY. In single-column indexes the options are indeed redundant, but in multicolumn indexes they can be useful. Consider a two-column index on (x, y): this can satisfy ORDER BY x, y if we scan forward, or ORDER BY x DESC, y DESC if we scan backward. But it might be that the application frequently needs to use ORDER BY x ASC, y DESC. There is no way to get that ordering from a plain index, but it is possible if the index is defined as (x ASC, y DESC) or (x DESC, y ASC).
Obviously, indexes with non-default sort orderings are a fairly specialized feature, but sometimes they can produce tremendous speedups for certain queries. Whether it's worth maintaining such an index depends on how often you use queries that require a special sort ordering.