Your system's first boot on its own power is what electrical engineers call the “smoke test”.
If you did a default installation, the first thing you should see when you boot the system is the menu of the
grub bootloader. The first choices in the menu will be for your new Debian system. If you had any other operating systems on your computer (like Windows) that were detected by the installation system, those will be listed lower down in the menu.
If the system fails to start up correctly, don't panic. If the installation was successful, chances are good that there is only a relatively minor problem that is preventing the system from booting Debian. In most cases such problems can be fixed without having to repeat the installation. One available option to fix boot problems is to use the installer's built-in rescue mode (see Section 8.6, “Recovering a Broken System”).
If you are new to Debian and Linux, you may need some help from more experienced users. For direct on-line help you can try the IRC channels #debian or #debian-boot on the OFTC network. Alternatively you can contact the debian-user mailing list. You can also file an installation report as described in Section 5.4.7, “Submitting Installation Reports”. Please make sure that you describe your problem clearly and include any messages that are displayed and may help others to diagnose the issue.
If you had any other operating systems on your computer that were not detected or not detected correctly, please file an installation report.