Working of indexes in PostgreSQL
This answer is about (default) B-tree indexes. See this later, related answer about GiST, GIN etc.:

Why is a GiST index used for filtering on non-leading column?


Here are the results of querying a table on the second column of a multicolumn index.
The effects are easy to reproduce for anybody. Try it at home.
I tested with PostgreSQL 9.0.5 on Debian using a medium sized table of a real-life database with 23322 rows. It implements the n:m relationship between the tables adr (address) and att (attribute), but that's not relevant here. Simplified schema:
CREATE TABLE adratt (
  adratt_id serial PRIMARY KEY
, adr_id    integer NOT NULL
, att_id    integer NOT NULL
, log_up    timestamp NOT NULL DEFAULT (now()::timestamp)
, CONSTRAINT adratt_uni UNIQUE (adr_id, att_id)
);

The UNIQUE constraint effectively implements a unique index. I repeated the test with a plain index to be sure and got identical results as expected.
CREATE INDEX adratt_idx ON adratt(adr_id, att_id);

The table is clustered on the adratt_uni index and before the test I ran:
CLUSTER adratt;
ANALYZE adratt;

Sequential scans for queries on (adr_id, att_id) are as fast as they can possibly be. The multicolumn index can still be used for a query condition on the second index column alone.
I ran the queries a couple of times to populate the cache and the picked the best of ten runs to get comparable results.
1. Query using both columns
SELECT *
FROM   adratt
WHERE  att_id = 90
AND    adr_id = 10;

 adratt_id | adr_id | att_id |       log_up
-----------+--------+--------+---------------------
       123 |     10 |     90 | 2008-07-29 09:35:54
(1 row)

Output of EXPLAIN ANALYZE:
Index Scan using adratt_uni on adratt  (cost=0.00..3.48 rows=1 width=20) (actual time=0.022..0.025 rows=1 loops=1)
  Index Cond: ((adr_id = 10) AND (att_id = 90))
Total runtime: 0.067 ms

2. Query using first column
SELECT * FROM adratt WHERE adr_id = 10;

 adratt_id | adr_id | att_id |       log_up
-----------+--------+--------+---------------------
       126 |     10 |     10 | 2008-07-29 09:35:54
       125 |     10 |     13 | 2008-07-29 09:35:54
      4711 |     10 |     21 | 2008-07-29 09:35:54
     29322 |     10 |     22 | 2011-06-06 15:50:38
     29321 |     10 |     30 | 2011-06-06 15:47:17
       124 |     10 |     62 | 2008-07-29 09:35:54
     21913 |     10 |     78 | 2008-07-29 09:35:54
       123 |     10 |     90 | 2008-07-29 09:35:54
     28352 |     10 |    106 | 2010-11-22 12:37:50
(9 rows)

Output of EXPLAIN ANALYZE:
Index Scan using adratt_uni on adratt  (cost=0.00..8.23 rows=9 width=20) (actual time=0.007..0.023 rows=9 loops=1)
  Index Cond: (adr_id = 10)
Total runtime: 0.058 ms

3. Query using second column
SELECT * FROM adratt WHERE att_id = 90;

 adratt_id | adr_id | att_id |       log_up
-----------+--------+--------+---------------------
       123 |     10 |     90 | 2008-07-29 09:35:54
       180 |     39 |     90 | 2008-08-29 15:46:07
...
(83 rows)

Output of EXPLAIN ANALYZE:
Index Scan using adratt_uni on adratt  (cost=0.00..818.51 rows=83 width=20) (actual time=0.014..0.694 rows=83 loops=1)
  Index Cond: (att_id = 90)
Total runtime: 0.849 ms


4. Disable indexscan &amp; bitmapscan
SET enable_indexscan = off;
SELECT * FROM adratt WHERE att_id = 90;

Output of EXPLAIN ANALYZE:
Bitmap Heap Scan on adratt  (cost=779.94..854.74 rows=83 width=20) (actual time=0.558..0.743 rows=83 loops=1)
  Recheck Cond: (att_id = 90)
  -&gt;  Bitmap Index Scan on adratt_uni  (cost=0.00..779.86 rows=83 width=0) (actual time=0.544..0.544 rows=83 loops=1)
        Index Cond: (att_id = 90)
Total runtime: 0.894 ms


SET enable_bitmapscan = off;
SELECT * FROM adratt WHERE att_id = 90;

Output of EXPLAIN ANALYZE:
Seq Scan on adratt  (cost=0.00..1323.10 rows=83 width=20) (actual time=0.009..2.429 rows=83 loops=1)
  Filter: (att_id = 90)
Total runtime: 2.680 ms

Conclusion
As expected, the multi-column index is used for a query on the second column alone.
As expected, it is less efficient, but the query is still 3x faster than without the index.
After disabling index scans, the query planner chooses a bitmap heap scan, which performs almost as fast. Only after disabling that, too, it falls back to a sequential scan.
See other answer for the original quote from the manual.
Updates since Postgres 9.0
Everything still basically true in Postgres 13. Most notable changes:

index-only scans in Postgres 9.2
True covering indexes with the INCLUDE keyword in Postgres 11
Multiple performance and space improvements (especially for multicolumn indexes) in Postgres 12
Deduplication in Postgres 13.

All in favor of index performance. (Sequential scans got faster, too, though.)