I have a dell laptop running by Windows XP. I turned it off on a
stand-by mode before I left it. There are two user, admin and another.
The other user doesn't need a password to get in. Now, how could I
know:

1- whether my computer was turned on during the period that I was not with it.
2- and if turned on, what of kind of things have been done on it.

Use the search files or folders utility to do it.  You can tell it to
search for files that were modifed after a specific time.  Use the
approximate time period when you suspect the computer may have been
used.

This will give you a reasonable idea whether or not the computer was
used and what they may have been doing.

Using the above method will not be strictly accurate, as soon as you
turn on your pc it writes to files and changes dates on files.

Turning it on will probably overwrite the same files as the ones the
user would have accessed when bringing the computer out of stand by
mode.

You best option is to check the Event Viewer

START -> Settings (or Control panel) -> control panel if not already
done -> Administrative tools -> Event viewer.

now try to find the time you think the computer was accessed and
search for any "Process was started" or more specifically search in
the System Log for any Information type entries under the source
"eventlog". Open the event and if it says information close to "The
Event log service was stopped." This is when your system went into
stand-by. Check the next newer event, if it says something such  as
"The Event log service was started." and the time this occurred is the
same time as you suspect your computer to have been used you know it
was turned on and someone tried to use it.

Ideally to get all this type of information you should be logging "log
on" or "Priviledged Access" attempts within your security logs.

--Keystroke-ga

perfect answer keystroke...

Way to GO