There are Office365 plans that allow you to download and install local versions of the full Office 2016 suite. In any event, most Office365 plans include Office Web Apps, which allows you to work with documents, spreadsheets, PowerPoint, etc. in a web browser.
For "home users", all the Office 365 plans come with the ability to locally install the desktop versions of the Office apps. The only thing that various is how many computers you can install them on, how many users your subscription supports (mainly for things like OneDrive storage space and associated email accounts), and whether or not it also includes the ability to install on tablets and/or phones. The web apps are free to use for home users, no subscription needed.
For "business users", then the lowest cost plan (currently $5 per month) is online apps only. The other two options allow the desktop apps to be installed on up to 5 computers per paid user (i.e. cannot pay for one user and then install the apps on 5 computers that 5 different people use). And as a business, you can mix and match plans (i.e. the lowest cost plan for some employees that do not need locally install desktop apps and higher level plans for those users who do need locally installed desktop apps).
OneDrive for Business is accessible from a web browser anywhere; the client app can also be installed on a computer.
There is also a OneDrive client for iOS, but not sure if it works with the business accounts (it does with the personal accounts...I have a personal account). It did not work with the business accounts for a while (I had a business account when working at a past company about 2 years ago).