If I know the PID number of a process, how can I get its name?

On all POSIX-compliant systems, and with Linux, you can use ps:

ps -p 1337 -o comm=

Here, the process is selected by its PID with -p. The -o option specifies the output format, comm meaning the command name.

For the full command, not just the name of the program, use:

ps -p 1337 -o command

See also: ps – The Open Group Base Specifications Issue 6

You can find the process name or the command used by the process-id or pid from

/proc/<pid>/cmdline

by doing

cat /proc/<pid>/cmdline

Here pid is the pid for which you want to find the name
For example:

 # ps aux

   ................
   ................
   user  2480  0.0  1.2 119100 12728 pts/0  Sl   22:42   0:01 gnome-terminal
   ................
   ................

To find the process name used by pid 2480 you use can

# cat /proc/2480/cmdline 

 gnome-terminal

To get the path of of the program using a certain pid you can use:

ps ax|egrep "^ [PID]"

alternatively you can use:

ps -a [PID]

Or also:

readlink /proc/[PID]/exe