The cloud is a generic term for services that traditionally were local on the computer being in the cloud instead. There is no hard and fast definition, just examples (and a lot of marketing).
File services like DropBox supply file systems in the cloud. You access those files either using a browser or an app that you load on the computer. There are tones of file services.
Businesses often use cloud sourced computing. Instead of running their own servers they rent time on big server farms. This has started to trickle down to end users with services like Google Docs, CloudOn, OnLive, and Microsoft 365; where you're tools and files are all in the cloud.
Apple's iCloud is a hodgepodge of features mostly centered around keeping all your information and devices synced and backed up. It has some limited file storage ability (depending on the app) and a few online tools; but they are focused towards it primary purpose, not meant to be the same type of solutions as DropBox or or Microsoft 365. At least not yet. There are indications Apple intends to make iCloud far more than it is now; step by seductive step.
The good news is you can use many of these services. You are not limited to iCloud.
So, to answer your more specific question. You get an app that can save your Power Point to an online service like DropBox. At school you download that Power Point using a browser (from your DropBox account) and use it as you usually would. If you have the option of downloading the DropBox app on the school computer (unlikely) it would already be there.
Or if you want to use Keynote, Box.com would probably work better. You can use the WebDAV export option to export files directly to it. Most of the other office suite apps have DropBox support built in.
You can find lots of opinions on the best app by searching the forum, and tons of reviews in Google. The most common are the iWorks apps (Pages, Numbers, Keynote), Office HD, QuickOffice, and Documents to Go.
At any rate, there are more and more services billing themselves as 'cloud' services every day. You're better off concentrating on the specifics of each product, and how it might fit into what you need to do.