There are a few different ways I have students hand in digital assignments. For larger projects (video and audio files mostly) the students can just save them in their profile folders or on our school media server. However if you don't have this set up, or if you want to be able to do marking at home, there are a few other ways.
USB flash drives: either student or school provided, students can save their work to individual flash drives, or have multiple students save their files on a single flash drive.
Email: I usually have students email their completed assignments to me, since we have gmail accounts set up for students and teachers with about 7 GB of storage space each. I can then mark the assignments on any computer with Internet access.
TurnItIn.com: our district subscribes to this website which provides originality checking and online marking, as well as opportunities for peer review.
Learning Management Systems: online systems can be set up for assignment submission as well as peer interaction et ceteras using something like Moodle or Ning.
I'm sure there are other ways that teachers are marking digital assignments, but these are just a few I've experimented with. Feel free to comment on some of the systems you've used or seen in use.