What command should I run to find out: who I am in the database and my priveleges ?

Thanks

i dont know how to find out "who you are". That should be known when you log in i guess , But the permissions database is the mysql database. It has many levels of permissions though. User level convering all databases, database level, table level and even record level. You'd have to run some SQL commands to get the permissions you have. I think the only user that has permission to this database however is the root user.

Other than that, I'm not sure if a user can obtain his/her permissions simply because they are kept in the mysql database.

You can use the command:
phpMyAdmin is much easier if you have a webserver though.

To see who you are, use:

select user();

To see what database you are connected to:

select database();

That's what my database says, but how can I add new user?

mysql> select user();
+----------------+
| user() |
+----------------+
| mike@localhost |
+----------------+
1 row in set (0.07 sec)

mysql> select database();
+------------+
| database() |
+------------+
| |
+------------+
1 row in set (0.00 sec)

Mysql Grant statement:
http://www.mysql.com/doc/en/GRANT.html

Solve the problem.. Need more help:

#1 MySQL is case sensitive or not ? When I change databases "use databasename" it is case sensitive, but when I create, drop, select it's not..
#2 How can I change host in the MYSQL ?

Thanks

You gave most of the answer already.
Names of databases and tables are case-sensitive.
Names of columns, and the SQL keywords (like selEct, fRom...) ar not.
With the -h (or --host) commandline option of mysql.

Is it possible to back up MySQL databases ?

Sure. See "man mysqldump".
You may also like to read about starting the mysql server with the --log option. You can use this in combination with the mysqldump options "--lock-tables" and "--flush-logs" to be able to recover a database passed the latest mysqldump back-up. Make sure the log is written to a different disk or even a different computer (i.e. using NFS), or make very regluar backups of the update log.