Things you need to watch out for when using a drill press are long hair and necklaces. Since your head is close to the spinning drill chuck, loose hair is much more likely to get sucked into the machine than a hand drill. A drill press will not stop running when your hair gets pulled in.
The other thing that injures people is when the bit grabs the material and either sends it flying or grabs it and starts spinning it like a finger smashing, wrist-breaking machine. It will keep going until you hit the power switch or the material eventually goes flying across the room. This is why it is important to either clamp your material down, or make sure it’s big enough that you can hold on to it.