Copy and Paste Rows into Same SQL Table with Different Values

Insert Into Rooms
     Select col1, col2, col3, 'Spring' as Semester -- select each column in order except 'Semester', pass it in literally as 'Spring'
     From rooms where
     Semester = 'Fall'

Well if you're just trying to do this inside Sql Server Management Studio you could copy the table, run an Update command and set the semester to spring on the cloned table, then use the wizard to append the data from the cloned table to the existing table.

If you know a programming language you could pull all of the data, modify the semester, then insert the data into the existing table.

Note: The other answers are a much better way of achieving this.

INSERT INTO rooms
(roomname, current_occupancy, max_occupancy, semester)
SELECT roomname, current_occupancy, max_occupancy,'spring'
FROM rooms
WHERE [semester]='fall'

(assuming names for your room and occupancy columns)

Use a temp table to make it simple no matter how many columns are involved;

SELECT * INTO #ROOMS FROM ROOMS;
UPDATE #ROOMS SET SEMESTER='spring';
INSERT INTO ROOMS SELECT * FROM #ROOMS;