Dropbox introduced a new feature “Links” which allow users to share their dropbox files with others. Not only files in public folder but you can now share any file you want. This feature works with every device you are using dropbox on, check this page to know sharing instructions for your particular device.
Check the following image for instruction on how to use this feature at dropbox.com. Click the image to enlarge.
You can share individual files as well as a complete folder. The other person you are sharing with need not be a user of dropbox, you can share files with anyone. Just send the link you get from dropbox to the other person. And it works just like other files on the web, it will either open up directly inside your browser or get downloaded to your device.