The grandaddy of all process monitors is top, and many system monitoring tools are called top. For example, there's iotop to watch disk I/O, atop for a bunch of system resources, powertop for power consumption.
If you want more detailed information, it's not tracked by default. To watch what a particular process is doing, call strace on it. For example, if you're only interested in filesystem accesses:
strace -s9999 -efile command_name # trace a program during its whole execution
strace -s9999 -efile -p1234 # trace a running program with the given PID
strace is specific to Linux, but other systems have a similar tool: truss on Solaris, ktrace or dtrace under *BSD, etc.
To watch what's happening to a particular file or in a particular directory or directory tree, use the inotify facility.
inotifywait -m .
Again, the facility is specific to Linux, but most other unices have a similar system, e.g. kqueue under *BSD, and FAM (originally from SGI but now available as an API on many systems).
To watch all the system calls under Linux, you can use the audit subsystem. It's relatively recent and there's not much literature on the topic; search for auditctl or read the auditctl man page. There are a couple of examples on this site: tracking file accesses, tracking process execution.