Dropbox is a SaaS (Software as a Service) product which allows you to store files remotely, access them from anywhere, share files, and sync a local directory to your offsite file store.
This is a great feature to have, but you might be wary of letting a third-party service keep your files. You might have privacy concerns, or you might simply think that it is unwise.
No problem — you can set up a similar file-syncing service on your own server with ownCloud.
You can set up your own home or office computer to act as a web server, or set it up on your own VPS hosting plan (or dedicated hosting if you need it).