Unfortunately, there can be many reasons for the problem. Most arise when the user selects the "upgrade" on the 10.5 installer rather than "archive and install" or a total wipe and install. This has been a problem for many who used the upgrade method. You did not mention how you installed nor what machine you have.
You also did not mention is you ran disk utility to check, and repair if necessary, your hard drive before installing the upgrade.
As a first check, try holding down the Shift key when you boot - starting the machine in "safe mode". If that works, you can try Apple menu/System prefs/System/startup disk. Select the hard drive, eject the installer disk and see what happens. If still stuck, try booting inot safe made again and check System prefs/Accounts/startup items. highlight each and hit the - button. See if you can then reboot ok.
If you still can't boot directly from the hard drive, try booting in safe mode again and go to hard drive.applications/utilities and launch disk utility. Check the hard drive for bad things and repair permissions. If the hard drive reports OK, try a normal reboot. If Hard drive problems are reported, boot form the 10.5 installer disk and open up utilities/disk utility and repair the drive.
You can also try resetting pram by holding down the command, option, p, r keys at boot. Hold them down until you hear the startup chime a second time.
All of this can drive you nuts. In many cases it's easier to put in your 10.3 disk, hit the Option button and do an archive and install. You might get a lot of warnings about replacing a new file. Choose replace. Boot normally and see if you're ok. If yes use software update to get all updates from Apple. Make sure you're ok. Then insert the 10.5 disk, click options and do an archive and install.
If you're fully backed up to an outboard disk, etc. you can also just use the 10.5 disk to do a total wipe and install.
Using OSX 10.3.9 on a G4 & G5