That is because you are doing so called cartesian product. It means that you will have the number of recods in table meetings times the number of records in table networkuni records returned back from the server for this select.
I am not sure, what exactly you need. But if the two tables has to be joined, then you have to join them like:
SELECT
m.meetingid As meetingid,
TRIM(m.title) As title,
TRIM(m.description) As description,
DATE_FORMAT(m.date_start, '%d-%m-%Y') As date_start,
DATE_FORMAT(m.date_end, '%d-%m-%Y') As date_end,
n.networkuniid As networkuniid,
TRIM(n.title) As title,
TRIM(n.text) As text,
DATE_FORMAT(n.date_start, '%d-%m-%Y') As date_start,
DATE_FORMAT(n.date_end, '%d-%m-%Y') As date_end
FROM
MEETINGS as m, NETWORKUNI as n
WHERE
date_start >= now() AND
m.meetingid = n.meetingid // !!!!!!!!
ORDER BY date_start DESC LIMIT 1
if you need only to combine the results then you can use UNION clause or you have to run two select statements and then solve the order.
Z.