What first aid can you give to somone who's got one of their fingers

[Laura]

Pressure!!! tie something around the hand until you can get them to the ER to prevent massive bleeding...

[LikeAShadow]

completely chopped off? Like stopping the bleeding.

[azure]

Agreed, the most important thing is to stop the bleeding. This means pressure (tying the wound up with something), also if possible elevate and immobilise the injured arm (e.g. a sling). Keep the person still and calm, preventing their blood pressure from jumping will slow the bleeding also.
Remember that if there is a lot of trauma or blood loss, the person may also go into shock, in which case, keep them warm and calm, lie them down and elevate the feet, and loosen any tight clothing or jewellery.

If you still have the severed digit and it's not completely mangled then wrap it in plastic and bring it to the ER. Put it on ice too if you can but don't forget to plastic-wrap first or it will be frostbitten!

Edit: Yes, always use a sterile dressing first if possible. However, if you are in a situation where you don't have anything sterile (i.e. no first aid kit), then bandaging with whatever you have is better than nothing.