There are at least three solutions to this problem.
The first one is to find an updated version of BIOS for your motherboard. It might allow booting from CD drive and thus solve the problem.
The second one is to use a distribution that still offers boot floppies. Most modern distributions rely on using a bootable live CD for installation. These suck for installation on an old computer as they usually cannot boot from CD and they don't have enough RAM to run an installer based on a live system. At least Debian and Slackware still offer floppies that can be used for getting the installation start.
If the old computer is connected to network, I would suggest installing the system over ftp. If you cannot install over network, you still need a CD for installation. At least my old Pentium has difficulties in reading rewritable CD-ROMs. If yours cannot read the installation CD, try to write another on a CD-ROM, not on a rewritable CD-RW.
The third solution is using Smart Boot Manager for booting the computer. Smart Boot Manager allows you then to continue booting from your CD drive.
I have used both the second and third solution for installing several distros on my Toshiba Satellite 200 that does not have a working network connection. Thus I need a distribution that has a reasonable collection of packages available on a small set of CDs. In practice this has meant the first three CDs of Debian or the full set of Slackware on three CDs.
You can download the image file for Smart Boot Manager floppy disk from here. After downloading the image, you have to make a boot floppy from it. For this, you need to use the command dd:
dd if=sbm.img of=/dev/fd0The syntax of dd is not very easy to remember unless one understands that the option if is shorthand for input file and of for output file. Thus the command above writes the image file sbm.img to device fd0 (floppy drive).
After writing the image to a floppy, boot the computer with Smart Boot Manager in the floppy drive. SBM allows you to continue booting the operating system from either hard disk or CD drive. Now you are ready to continue installation from the bootable CD.