Do search engines (google) consider a sub domain to be a separate domain or is it just like any other folder/ directory under the domain? Does the page rank from the root domain carry over to the sub domain?

The page rank would be the same for the main domain. In most set ups:

subdomain.domain.com = domain.com/subdomain

So consequently, googlebots will reflect page rank on the main domain, and crawl the content for the subdomain as it would for any other folder.

Thanks for the speedy reply!

Can you set up a sub domain like a separately hosted domain? (It is on the same server but has its own ftp access - ID & password)

So, is it possible to buy just a sub domain from the owner of a particular domain and use it independantly like my own site but benefit from the existing page rank?

Thats correct, you can set up an ftp account for a particular folder on a domain that only certain users can access. However, remember page rank takes into account all user pages and links, so just as you can take advantage of page rank, your content can also decrease it.

Wow!
This is awesome!
What if (for the obvious ownership of content) I hosted the sub domain on my own server? (got the owner to point only the sub to my DNS)

Would I still benefit from his root domain page rank or will the googlebots see this as an entirely separate domain and I will have to start from PR0?

At the point where the folder was hosted on its own server, it would lose its association to the page rank. Google has a well respected algorithm for page rank, and one that reacts to fairly minute changes quite quickly.

If you're going to do the subdomain thing, I would suggest that the content be complimentary or have enough interest to grab page rank on its own; although initially you might have some success, like a parents success projects onto the child, but after a time, your content will have to stand on its own.

Most people believe search engines view a subdomain as a seperate domain.