Most of the "cloud" services that have an Android app should work for you (e.g. Box - app, Dropbox - app, Google Drive - app, SugarSync - app, etc.)
However, keep in mind that most such 3rd-party services store your data on their own servers. If you are conscious about sensitivity of your data and want more control over how and where it's stored, the off-the-shelf services may not be the solution for you. You'll need to set up your own FTP/WebDAV/etc. server, and make it available online by modifying your network's firewall rules to allow outside connectivity. If you have a dynamic IP address from your ISP, you'll also want to register with a free service like No-IP which gives your server a static URL and little Windows/Mac/Unix utility to keep the server's IP address in sync, so that it's always accessible.
You can then use a different kind of a sync app which allows for specifying address/type of the server to sync to. Sachin's suggestion FolderSync will work perfectly for that.