Andrewst spent a lot of time putting together a sticky called 'Guidance & Resources for Posters' precisely for people like you. Read the sticky, then if you get stuck on any of the technical apsects of backups come back & ask for help.
Although it is NOT and can NOT be used instead of a proper backup, I believe that - in such a case ("teacher wants to see what I've done so far") - export might do the job quite well.
"A small database" is probably only one schema and it is quite simple to perform an export, put these several MB on USB memory stick, give it to the teacher, let him import my schema and do whatever he/she wants to.
If that's acceptable, read something more about the original Export and Import utilities in Oracle 10g documentation; if you use another database version, check its documentation at http://tahiti.oracle.com.