An HDD install is an install with the cd?
Both installs can be done from the CD. The CD contains the system that you want to copy to your computer. A Frugal install means the system is compressed on the HDD and can't be changed directly, while a plain HDD install just copies the files directly, so more space is used, but you can modify things easilly.
What is a partition?
A partition is basically just an area of the HDD that the Operating System is allowed to read and write data to. If you have more than one partition, you can treat them like different drives all togeather and use them for different jobs. With DSL, we only really need one partition which covers the whole drive (which is what I described previously), for Knoppix you would need another one to act as extra memory in case you ran out.
Is Knoppix small like DSL? What makes it better then DSL?
DSL is designed to run on old computers from the late 1990s era, and also to be very small in size. Due to this, it is limited to an old set of core software which is now not compatible with a lot of newer programs (like many of the latest web browsers). Knoppix supports newer software, including some packages that may make the system easier for you to use. I don't know what you want to do with your computer, that's why I suggested you try DSL running from the CD and then decide whether it does what you want before installing it permanently. If it doesn't suit you, Knoppix is another option that might.
If I choose Knoppix, I would follow the same instructions you posted here?
You could use a similar process to that I described earlier for installing Knoppix, but you would have to create two partitions (one for the Operating System files, another "swap" partition to act as extra memory) of different sizes on the drive.