Thank you. I did manage to make this work.
As I mentioned, I think at the top of the thread, the entire exercise was to verify that the name of the second database needs to be ONLY in front of the FROM table. The rest of the SQL Select can simply use the column names as is.

>When you concatenate strings you must carry about single quotes around strings:
>

>set @sql = 'select * from ' + @cSecondDb + '.dbo.mytable' + ' WHERE  Category = ''' + @Category+''''
>

>
>>Yes. I am getting an error:
>>Incorrect syntax near 'F'.
>>
>>>Were you using something like the following?
>>>

>>>use [FirstSQLDatabase];
>>>go 
>>>declare @cSecondDb as Varchar(100);
>>>declare @sql as varchar(max);
>>>declare @Category as Varchar(25);
>>>set @Category = '1009F';
>>>set @cSecondDb = 'SecondSQLDatabase';
>>>set @sql = 'select * from ' + @cSecondDb + '.dbo.mytable' + ' WHERE  Category = ' + @Category
>>>exec (@sql);
>>>

>>>
>>>>I know that I am missing something simple. But how do I add a WHERE clause to the dynamic SQL?
>>>>For example, say I want to add to the SQL Select below:
>>>>

>>>>declare @Category as Varchar(25);
>>>>set @Category = '1009F';
>>>>' WHERE  Category = @Category'
>>>>

>>>>I get error no matter what I do with the @Category.
>>>>
>>>>>Hi Dmitry,
>>>>>
>>>>>Use dynamic SQL, something like this:
>>>>>

>>>>>use [FirstSQLDatabase];
>>>>>go 
>>>>>declare @cSecondDb as Varchar(100);
>>>>>declare @sql as varchar(max);
>>>>>set @cSecondDb = 'SecondSQLDatabase';
>>>>>set @sql = 'select * from ' + @cSecondDb + '.dbo.mytable';
>>>>>exec (@sql);
>>>>>

>>>>>
>>>>>and replace dbo with the appropriate schema name if it's not dbo.
>>>>>
>>>>>
>>>>>>Hi,
>>>>>>
>>>>>>I am testing the SQL Select with a name of the database not current.
>>>>>>This case works:
>>>>>>

>>>>>>use [FirstSQLDatabase]
>>>>>>go
>>>>>>select * from SecondSQLDatabase..mytable where 1=1
>>>>>>

>>>>>>
>>>>>>Now I want to put the 'SecondSQLDatabase' into a variable. Here is how I do it:
>>>>>>

>>>>>>use [FirstSQLDatabase]
>>>>>>go 
>>>>>>declare @cSecondDb as Varchar(100)
>>>>>>set @cSecondDb = 'SecondSQLDatabase'
>>>>>>select * from @cSecondDb..mytable where 1=1    -- this is the line that causes the syntax error.
>>>>>>

>>>>>>
>>>>>>The above produces syntax error.
>>>>>>
>>>>>>What am I missing?
>>>>>>TIA