With cloud-based file storage and syncing , you're actually renting a tiny portion of a server owned by a service provider. You still have a PC, or five, but you install some code that watches for new files to appear in a designated folder. When they do, the code initiates an automatic sync, which uploads the new data over your Internet connection to your little piece of the provider's servers and then downloads it to all connected PCs. Most providers also have developed mobile apps, so you can access these files on your tablet and phone.
The great thing about cloud storage is that it backs up your data outside of your home or office, which is especially handy if, say, a house fire consumes your PC.
Dropbox and SugarSync are two popular file storage and syncing providers, but you can find dozens of solutions, including some hosted by big-name technology brands: Amazon (Cloud Drive), Google (Google Drive) and Microsoft (SkyDrive). Most provide some storage -- 5 gigabytes, say -- for free and then charge a fee based on how much additional storage you use.