What's the best way to send a signal to all members of a process group?

You don't say if the tree you want to kill is a single process group. (This is often the case if the tree is the result of forking from a server start or a shell command line.) You can discover process groups using GNU ps as follows:

 ps x -o  "%p %r %y %x %c "

If it is a process group you want to kill, just use the kill(1) command but instead of giving it a process number, give it the negation of the group number. For example to kill every process in group 5112, use kill -TERM -- -5112.

Kill all the processes belonging to the same process tree using the Process Group ID (PGID)

kill -- -$PGID Use default signal (TERM = 15)
kill -9 -$PGID Use the signal KILL (9)

You can retrieve the PGID from any Process-ID (PID) of the same process tree

kill -- -$(ps -o pgid= $PID | grep -o '[0-9]*') (signal TERM)
kill -9 -$(ps -o pgid= $PID | grep -o '[0-9]*') (signal KILL)

^{Special thanks to tanager and Speakus for contributions on $PID remaining spaces and OSX compatibility.}

Explanation

kill -9 -"$PGID" => Send signal 9 (KILL) to all child and grandchild...
PGID=$(ps opgid= "$PID") => Retrieve the Process-Group-ID from any Process-ID of the tree, not only the Process-Parent-ID. A variation of ps opgid= $PID is ps -o pgid --no-headers $PID where pgid can be replaced by pgrp.
But:
- ps inserts leading spaces when PID is less than five digits and right aligned as noticed by tanager. You can use:
  PGID=$(ps opgid= "$PID" | tr -d ' ')
- ps from OSX always print the header, therefore Speakus proposes:
  PGID="$( ps -o pgid "$PID" | grep [0-9] | tr -d ' ' )"
grep -o [0-9]* prints successive digits only (does not print spaces or alphabetical headers).

Further command lines

PGID=$(ps -o pgid= $PID | grep -o [0-9]*)
kill -TERM -"$PGID"  # kill -15
kill -INT  -"$PGID"  # correspond to [CRTL+C] from keyboard
kill -QUIT -"$PGID"  # correspond to [CRTL+\] from keyboard
kill -CONT -"$PGID"  # restart a stopped process (above signals do not kill it)
sleep 2              # wait terminate process (more time if required)
kill -KILL -"$PGID"  # kill -9 if it does not intercept signals (or buggy)

Limitation

As noticed by davide and Hubert Kario, when kill is invoked by a process belonging to the same tree, kill risks to kill itself before terminating the whole tree killing.
Therefore, be sure to run the command using a process having a different Process-Group-ID.

Long story

> cat run-many-processes.sh
#!/bin/sh
echo "ProcessID=$$ begins ($0)"
./child.sh background &
./child.sh foreground
echo "ProcessID=$$ ends ($0)"

> cat child.sh
#!/bin/sh
echo "ProcessID=$$ begins ($0)"
./grandchild.sh background &
./grandchild.sh foreground
echo "ProcessID=$$ ends ($0)"

> cat grandchild.sh
#!/bin/sh
echo "ProcessID=$$ begins ($0)"
sleep 9999
echo "ProcessID=$$ ends ($0)"

Run the process tree in background using '&'

> ./run-many-processes.sh &    
ProcessID=28957 begins (./run-many-processes.sh)
ProcessID=28959 begins (./child.sh)
ProcessID=28958 begins (./child.sh)
ProcessID=28960 begins (./grandchild.sh)
ProcessID=28961 begins (./grandchild.sh)
ProcessID=28962 begins (./grandchild.sh)
ProcessID=28963 begins (./grandchild.sh)

> PID=$!                    # get the Parent Process ID
> PGID=$(ps opgid= "$PID")  # get the Process Group ID

> ps fj
 PPID   PID  PGID   SID TTY      TPGID STAT   UID   TIME COMMAND
28348 28349 28349 28349 pts/3    28969 Ss   33021   0:00 -bash
28349 28957 28957 28349 pts/3    28969 S    33021   0:00  \_ /bin/sh ./run-many-processes.sh
28957 28958 28957 28349 pts/3    28969 S    33021   0:00  |   \_ /bin/sh ./child.sh background
28958 28961 28957 28349 pts/3    28969 S    33021   0:00  |   |   \_ /bin/sh ./grandchild.sh background
28961 28965 28957 28349 pts/3    28969 S    33021   0:00  |   |   |   \_ sleep 9999
28958 28963 28957 28349 pts/3    28969 S    33021   0:00  |   |   \_ /bin/sh ./grandchild.sh foreground
28963 28967 28957 28349 pts/3    28969 S    33021   0:00  |   |       \_ sleep 9999
28957 28959 28957 28349 pts/3    28969 S    33021   0:00  |   \_ /bin/sh ./child.sh foreground
28959 28960 28957 28349 pts/3    28969 S    33021   0:00  |       \_ /bin/sh ./grandchild.sh background
28960 28964 28957 28349 pts/3    28969 S    33021   0:00  |       |   \_ sleep 9999
28959 28962 28957 28349 pts/3    28969 S    33021   0:00  |       \_ /bin/sh ./grandchild.sh foreground
28962 28966 28957 28349 pts/3    28969 S    33021   0:00  |           \_ sleep 9999
28349 28969 28969 28349 pts/3    28969 R+   33021   0:00  \_ ps fj

The command pkill -P $PID does not kill the grandchild:

> pkill -P "$PID"
./run-many-processes.sh: line 4: 28958 Terminated              ./child.sh background
./run-many-processes.sh: line 4: 28959 Terminated              ./child.sh foreground
ProcessID=28957 ends (./run-many-processes.sh)
[1]+  Done                    ./run-many-processes.sh

> ps fj
 PPID   PID  PGID   SID TTY      TPGID STAT   UID   TIME COMMAND
28348 28349 28349 28349 pts/3    28987 Ss   33021   0:00 -bash
28349 28987 28987 28349 pts/3    28987 R+   33021   0:00  \_ ps fj
    1 28963 28957 28349 pts/3    28987 S    33021   0:00 /bin/sh ./grandchild.sh foreground
28963 28967 28957 28349 pts/3    28987 S    33021   0:00  \_ sleep 9999
    1 28962 28957 28349 pts/3    28987 S    33021   0:00 /bin/sh ./grandchild.sh foreground
28962 28966 28957 28349 pts/3    28987 S    33021   0:00  \_ sleep 9999
    1 28961 28957 28349 pts/3    28987 S    33021   0:00 /bin/sh ./grandchild.sh background
28961 28965 28957 28349 pts/3    28987 S    33021   0:00  \_ sleep 9999
    1 28960 28957 28349 pts/3    28987 S    33021   0:00 /bin/sh ./grandchild.sh background
28960 28964 28957 28349 pts/3    28987 S    33021   0:00  \_ sleep 9999

The command kill -- -$PGID kills all processes including the grandchild.

> kill --    -"$PGID"  # default signal is TERM (kill -15)
> kill -CONT -"$PGID"  # awake stopped processes
> kill -KILL -"$PGID"  # kill -9 to be sure

> ps fj
 PPID   PID  PGID   SID TTY      TPGID STAT   UID   TIME COMMAND
28348 28349 28349 28349 pts/3    29039 Ss   33021   0:00 -bash
28349 29039 29039 28349 pts/3    29039 R+   33021   0:00  \_ ps fj

Conclusion

I notice in this example PID and PGID are equal (28957).
This is why I originally thought kill -- -$PID was enough. But in the case the process is spawn within a Makefile the Process ID is different from the Group ID.

I think kill -- -$(ps -o pgid= $PID | grep -o [0-9]*) is the best simple trick to kill a whole process tree when called from a different Group ID (another process tree).

pkill -TERM -P 27888

This will kill all processes that have the parent process ID 27888.

Or more robust:

CPIDS=$(pgrep -P 27888); (sleep 33 && kill -KILL $CPIDS &); kill -TERM $CPIDS

which schedule killing 33 second later and politely ask processes to terminate.

See this answer for terminating all descendants.

To kill a process tree recursively, use killtree():

#!/bin/bash

killtree() {
    local _pid=$1
    local _sig=${2:--TERM}
    kill -stop ${_pid} # needed to stop quickly forking parent from producing children between child killing and parent killing
    for _child in $(ps -o pid --no-headers --ppid ${_pid}); do
        killtree ${_child} ${_sig}
    done
    kill -${_sig} ${_pid}
}

if [ $# -eq 0 -o $# -gt 2 ]; then
    echo "Usage: $(basename $0) <pid> [signal]"
    exit 1
fi

killtree $@

rkill command from pslist package sends given signal (or SIGTERM by default) to specified process and all its descendants:

rkill [-SIG] pid/name...