I don't have personal experience of depression, but I understand that there can be days when sufferers can barely make it out of bed, let alone out of the house to walk a dog. If your depression is serious and debilitating, choose a pet that needs less attention; remember that animals never have off days.
If your depression is mild, you could think about either a dog or cat. Nurturing something else and considering its needs will help to shift your focus away from your illness. Stroking animals is also therapeutic, so make sure you choose a tactile animal that likes human beings.
Pet shop owners say that rats are greatly underrated. They are very clean and can be trained; something that's fun and a focus for the time you spend together. They also are diurnal, so you'll see them during the day.
But first you should think about a safety net, perhaps asking a family member if he or she will help out on bad days. Put that into place before you commit.
Another idea is to volunteer to work with animals. This would allow you to test the water first, and to meet people and gain the experience and skills you'll need to look after a pet.