Error while creating trigger

First problem: Each DECLARE line is followed by a comma, indicating that the statement continues on the next line. Either remove the commas, replace them with semi-colons, or remove all but the first DECLARE:

declare @user_id int
declare @user_name varchar(50)
declare @password varchar(50)
declare @firstname varchar(50)
declare @lastname varchar(50)
declare @audit_Action varchar(100)

-- Or:
declare @user_id int;
declare @user_name varchar(50);
declare @password varchar(50);
declare @firstname varchar(50);
declare @lastname varchar(50);
declare @audit_Action varchar(100);

-- Or:
declare @user_id int,
        @user_name varchar(50),
        @password varchar(50),
        @firstname varchar(50),
        @lastname varchar(50),
        @audit_Action varchar(100)

Second problem: You've copied the DECLARE lines, and simply replaced the DECLARE keyword with SELECT. This is totally invalid syntax.

Third problem: Triggers can and do fire for multiple rows at the same time. It looks like your code is trying to process a single updated row, which isn't going to work.

Try something like this:

CREATE TRIGGER trg_update
ON dbo.registeruser
AFTER UPDATE
AS
BEGIN

declare @user_id int;
declare @user_name varchar(50);
declare @password varchar(50);
declare @firstname varchar(50);
declare @lastname varchar(50);
declare @audit_Action varchar(100);

    If update(user_id) Or update(user_name) Or update(password) Or update(firstname) Or update(lastname)
    BEGIN
        -- This doesn't seem to be used anywhere:
        -- SET @audit_action = 'Updated Record -- After Update Trigger.';

        INSERT INTO old_registeruser
        (
            user_id,
            user_name,
            password,
            firstname,
            lastname
        )
        SELECT
            user_id,
            user_name,
            password,
            firstname,
            lastname
        FROM
            deleted
        ;
    END;

    PRINT 'AFTER UPDATE Trigger fired.'
END

Other comments:

The fact that you have a password column suggests that you're storing passwords in plain text. That is a very bad idea - you should be storing a salted hash of the password.
Salted Password Hashing - Doing it Right[^]

Finally, why have you tagged this SQL question as "C#"?!