Trigger After Insert is preventing insertion
The trigger is part of the original transaction. Just because it happens after the original insert does not mean that everything doesn't get rolled back in the event of an error. A rollback does not just roll back the last thing that ran. Take this example:

BEGIN TRANSACTION;
INSERT dbo.table1 ...;
INSERT dbo.table2 ...;
ROLLBACK TRANSACTION;


Surely you don't expect the rollback to only affect the 2nd insert?
You might consider using TRY/CATCH or an INSTEAD OF trigger, which will allow you to handle other stuff without suffering from the errors. Or simply checking first 
Your use of declaring a bunch of variables, assigning them from a join, and then inserting seems to be both inefficient and problematic. What happens if more than one row matches the join? You know that if you have a multi-row operation, the trigger only fires once, so you'll only end up with one arbitrary row being inserted, right? Why not:

INSERT dbo.TBL_USER_LOGS(DEV_NID, LOG_DATE, USE_NID, LOG_MODE)
  SELECT DEV_NID, LOG_DATE, REL_ENROLLNUMBER, REL_MODE -- table prefixes?
FROM dbo.TBL_DEVICE -- always use dbo prefix! 
INNER JOIN dbo.TBL_READER_LOG 
ON DEV_ORIGINAL_NID = RDR_NID -- which table do these columns come from?
WHERE DEV_TYPE = 1; -- which table does this column come from?


As an aside, I notice that your query doesn't seem to take the inserted pseudo-table into account. Shouldn't the insert have something to do with the row(s) actually affected by the trigger?