If you go in the winter, pack lots of layers. Also shoes with good grip as the sidewalks can be slippery.
Photo Equipment: The scemery is lovely and so you should take the chance to capture a lot of photos
Camping/Beach/Outdoor Gear: ski/snowboard equipment.
Luggage and bags:
Bring a small backpack if you're planning on going for a day-hike. You will need to bring your own water and some food. The last place to get food on your way up is Morskie Oko (the glacial lake "Eye of the Sea").
Clothing/Shoes/Weather Gear: If you're just planning to go to Morskie Oko you can get away with sneakers or even very comfortable sandals (that's what I had). The bigger mountains are the real deal though, and you should bring your real-deal hiking boots to conquer them.
Toiletries and Medical Supplies: If you're staying in the wallet-friendly Dom Turysty PTTK as I recommended, you will need all the basic toiletries such as soap, and even a towel. They supply nothing but the bed and bathroom. You might want some flip flops for the shower too. It's pretty gross.
Camping/Beach/Outdoor Gear: Bring a Swiss army knife. This is the mountains and you will need it even if you're just coming to stay in a hotstel for the night. We had to go to the supermarket to get sandwitch building materials for our hike, and had to stal knifes from a cafeteria to cut up all the meat and cheese.
Miscellaneous: If you're staying for a few days, bring a plug for the sink so you can wash your socks and underwear.