U is an arbitrary measure that is specific for your particular protein and you need to define it. For example, 1 unit (U) of SalI nuclease is defined as the amount required to digest 1 ug of lambda DNA at 37 C in 1 hour (I think off the top of my head). Whilst 1 unit of cytochrome c reductase is the amount required to reduce 1 mol of cytochrome c at 25 C (I think) per minute. Obvioiusly this is also dependent on the volume being used, and so is often accompanied by a specific protocol under which the conditions can be controlled.
Not sure if the actual examples presented here are the official numbers, but you get the idea.
I would also add that "U/ml" may not be particularly informative more often activity is standardized to protein content to give a "specific activity suck as U/mg protein. This standardization accounts for differences in protein preparations and also is used to assess the purity of enzyme preparations. As to Jason's answer yes you can create your own definition of U but for many enzymes a standardized definition of U has been developed most often relating to the rate of production of the product or use of the substrate