My guess is one of these:
1. Phones usually do need drivers to allow data transfer. Even if it worked before apparently without (maybe the other computer downloaded automatically from internet), you should check the support site for your phone' maker and check for USB drivers.
2. Phone is not in computer transfer mode. If you computer has internal memory or memory card (SD, memory stick pro, etc), usually there is an option in the phone's configuration to allow USB access to that.
3. Would be useful if you could tell us the maker and model of your phone.