The last line assigns the result of the shell escaped ps | awk pipe to a shell variable called "pids" (this is indicated by the $( and ) pair).

Within the shell, the ps command lists all processes, with the two flags 'c' and 'u', which change the command name to be just the executable name, and display useful information too, and the '-U user' flag limits results to just the specified user (in case more than one user has processes running on the tty device).

The output is expected to look like this:

$ ps -cu -U taylor
USER     PID %CPU %MEM      VSZ    RSS  TT  STAT STARTED      TIME COMMAND
taylor   222   4.3 -3.9   329032  81652  ??  Ss   Thu08PM 108:00.19 WindowServer
taylor  1760   3.3 -0.4   202792   8792  ??  R    10:09AM   0:09.09 Terminal
taylor  1162   1.0 -1.2   269288  25800  ??  S    Sat08AM  29:42.57 iChat
taylor   714   0.4 -1.3   242220  27564  ??  S    Fri08AM  23:57.56 iTunes
taylor   287   0.0 -0.6   208284  13468  ??  S    Thu08PM   1:53.66 Dock

Now you can see that if we search for a specific pattern, we'll match all lines that are associated with a specific process name. This could be done with grep, but using awk is useful because we can then control which of the output fields is listed too (we only want the PID, field #2). That's what the second part of the pipe does.

The problem you're seeing is that your version of Unix isn't returning the same results for the ps -cu -Uuser command. That being the case, you're probably seeing output like this:

  PID  TTY       TIME COMMAND
 10361 ttyrb     0:00 sleep

pids=$(ps cu -U $user | awk "/ $1$/ { print \$1 }")