SELECT  m1.meetingID, m1.meetingStart, m1.meetingEnd, m2.meetingID
FROM    t_meeting m1, t_meeting m2
WHERE   (m2.meetingStart BETWEEN m1.meetingStart AND m1.meetingEnd
        OR m2.meetingEnd BETWEEN m1.meetingStart AND m1.meetingEnd)
        AND m1.meetingID <> m2.meetingID
This will select each pair twice.
If you want each pair to be selected just once, use:
SELECT  m1.meetingID, m1.meetingStart, m1.meetingEnd, m2.meetingID
FROM    t_meeting m1, t_meeting m2
WHERE   (m2.meetingStart BETWEEN m1.meetingStart AND m1.meetingEnd
        OR m2.meetingEnd BETWEEN m1.meetingStart AND m1.meetingEnd)
        AND m2.meetingID > m1.meetingID
Make sure you have indexes on meetingStart and meetingEnd for the query to work efficiently.
MySQL, however, will probably use INDEX MERGE to run this query, which is not very efficient in current implementation.
You also may try to use:
SELECT  m1.*, m2.*
FROM    (
        SELECT  m1.meetingID AS mid1, m2.meetingID AS mid2
        FROM    t_meeting m1, t_meeting m2
        WHERE   m2.meetingStart BETWEEN m1.meetingStart AND m1.meetingEnd
                AND m2.meetingID <> m1.meetingID
        UNION
        SELECT  m1.meetingID, m2.meetingID
        FROM    t_meeting m1, t_meeting m2
        WHERE   m2.meetingEnd BETWEEN m1.meetingStart AND m1.meetingEnd
                AND m2.meetingID <> m1.meetingID
        ) mo, t_meeting m1, t_meeting m2
WHERE   m1.meetingID = mid1
        AND m2.meetingID = mid2
, which is more complex but will most probably run a little bit faster.