Move a row from one table to another

Well, the answer to this is pretty simple. You can't UNION a DELETE and an INSERT. You can only UNION SELECTs. You'll just have to run the DELETE and INSERT as separate statements. Depending on your database system, you might be able to submit both of them in one call to the database if they're separated by a semicolon.

Also, it's more common to do the INSERT first. Typically, you'll want to copy some of the fields from one table to the other. So you would do something like:

insert
  into table2
     ( pkey
     , field1
     , field2
     , fieldN )
select pk
     , col1
     , 'foo2'
     , colN      
  from table1
 where pk = 937

Often, you would run these queries from some scripting language like Coldfusion, ASP, or PHP. It might be wise to check the status code of the INSERT to make sure it was successful before continuing with the DELETE.

delete
  from table1  
 where pk = 937

Yes, there's extra overhead in making two calls to the database instead of one, but it's minimal.