SQL Inner-join with 3 tables?

You can do the following (I guessed on table fields,etc)

SELECT s.studentname
    , s.studentid
    , s.studentdesc
    , h.hallname
FROM students s
INNER JOIN hallprefs hp
    on s.studentid = hp.studentid
INNER JOIN halls h
    on hp.hallid = h.hallid

Based on your request for multiple halls you could do it this way. You just join on your Hall table multiple times for each room pref id:

SELECT     s.StudentID
    , s.FName
    , s.LName
    , s.Gender
    , s.BirthDate
    , s.Email
    , r.HallPref1
    , h1.hallName as Pref1HallName
    , r.HallPref2 
    , h2.hallName as Pref2HallName
    , r.HallPref3
    , h3.hallName as Pref3HallName
FROM  dbo.StudentSignUp AS s 
INNER JOIN RoomSignUp.dbo.Incoming_Applications_Current AS r 
    ON s.StudentID = r.StudentID 
INNER JOIN HallData.dbo.Halls AS h1 
    ON r.HallPref1 = h1.HallID
INNER JOIN HallData.dbo.Halls AS h2
    ON r.HallPref2 = h2.HallID
INNER JOIN HallData.dbo.Halls AS h3
    ON r.HallPref3 = h3.HallID