How does Basic mode work?
Autorun Manager in basic mode lists all installed apps which are intended to start at boot, if you click on any of the apps you can disable it, that will kill it when it starts at boot time. Please note that on Froyo and newer android versions apps may restart themself, these apps will be marked as !!self restarter!! in the application list.
I have disabled an app but it is still started after boot/is restarted, why?
Some services are written to automatically restart themself if they are killed, these are detected by Autorun Manager and marked in application list on the bottom right as !!self restarter!! (this label appears only for disabled apps after reboot) you really have better leave these enabled.
How does Advanced mode work?
Advanced mode list all the applications which are registered for system intents and shows you what receivers listen to what intents. It also gives you the control over these, so you can enable and disable these. With the help of Advanced mode you gain real and full control.
Application says root access needed. What is it and how can i get it?
Root access means you have superuser (su) permissions on your phone (only su can modify some parts of the system). To achieve it you have to make some modifications on your phones software which may hurt your guarantee, on the other side it will make it available to install custom ROMs, do full system backups, tether your internet, install applications to sd, run AutoKiller (and many other su related apps), and much more (check this, this and this). The internet is full with descriptions how to root your phone, you shall just google for it, or start here. You can also check this wiki for more info.