Your system's first boot on its own power is what electrical engineers call the “smoke test”.
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 less common architectures like 32-bit MIPS (little-endian), your best option is to ask on the debian-mips mailing list. You can also file an installation report as described in Section 5.4.5, “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.