The command you are looking at is actually this:
[ ! -S "qdata/c$i/tm.ipc" ]
Although it looks like punctuation, [ is actually the name of a command, also called test; so the command can also be written like this:
test ! -S "qdata/c$i/tm.ipc"
Which in context would look like this:
if test ! -S "qdata/c$i/tm.ipc"; then
DOWN=true
fi
As the name suggests, its job is to test some attribute of a string, number, or file, and return 0 (which represents true in shell scripts) if the test passes, and 1 (which represents false) if it doesn’t.
Armed with this knowledge, you can run man test, and find the following explanations of the ! and -S arguments:
! EXPRESSION
EXPRESSION is false
and
-S FILE
FILE exists and is a socket
So test ! -S filename or [ ! -S filename ] can be read as “not is-socket filename”.
So the command is checking whether a “socket” (a special kind of file) exists with each name in the loop. The script uses this command as the argument to an if statement (which can take any command, not just [) and sets DOWN to true if any of them does not exist.