What does it mean by select 1 from table?

Question

select 1 from table will return the constant 1 for every row of the table. It's useful when you want to cheaply determine if record matches your where clause and/or join.

SELECT 1 FROM TABLE_NAME means, "Return 1 from the table". It is pretty unremarkable on its own, so normally it will be used with WHERE and often EXISTS (as @gbn notes, this is not necessarily best practice, it is, however, common enough to be noted, even if it isn't really meaningful (that said, I will use it because others use it and it is "more obvious" immediately. Of course, that might be a viscous chicken vs. egg issue, but I don't generally dwell)).

 SELECT * FROM TABLE1 T1 WHERE EXISTS (
     SELECT 1 FROM TABLE2 T2 WHERE T1.ID= T2.ID
 );

Basically, the above will return everything from table 1 which has a corresponding ID from table 2. (This is a contrived example, obviously, but I believe it conveys the idea. Personally, I would probably do the above as SELECT * FROM TABLE1 T1 WHERE ID IN (SELECT ID FROM TABLE2); as I view that as FAR more explicit to the reader unless there were a circumstantially compelling reason not to).

EDIT

There actually is one case which I forgot about until just now. In the case where you are trying to determine existence of a value in the database from an outside language, sometimes SELECT 1 FROM TABLE_NAME will be used. This does not offer significant benefit over selecting an individual column, but, depending on implementation, it may offer substantial gains over doing a SELECT *, simply because it is often the case that the more columns that the DB returns to a language, the larger the data structure, which in turn mean that more time will be taken.

If you mean something like

SELECT * FROM AnotherTable
  WHERE EXISTS (SELECT 1 FROM table WHERE...) 

then it's a myth that the 1 is better than

SELECT * FROM AnotherTable
  WHERE EXISTS (SELECT * FROM table WHERE...) 

The 1 or * in the EXISTS is ignored and you can write this as per Page 191 of the ANSI SQL 1992 Standard:

SELECT * FROM AnotherTable
  WHERE EXISTS (SELECT 1/0 FROM table WHERE...) 

it does what it says - it will always return the integer 1. It's used to check whether a record matching your where clause exists.

select 1 from table is used by some databases as a query to test a connection to see if it's alive, often used when retrieving or returning a connection to / from a connection pool.