Any old backpack will do. However, there are specialised cycling backpacks that have compartments for phone, wallet and keys, and have handy features like reflective strips for riding at night.
Pros: Easy to take with you - carry a little or carry a lot
Cons: Sweaty back, and uncomfortable if too full
Panniers sit on either side of your bike so you can balance the load or just take one. You will need to fit a rack either to the front or rear of your bike.
Pros: Comfortable, large capacity
Cons: Can affect handling if heavy
Most panniers and some specialist bike backpacks come with them, but you can get ‘one-size-fits-most’ options too. They not only keep your gear dry, they protect your bag, and their bright colours are good for visibility.
Pros: Cheap, keep your stuff dry, increase your visibility
Cons: Harder to get to your gear
Suitable for much larger cargo and not so useful for a daily commute. Also handy if you are riding to a destination (eg camping) and you can detach the trailer to go riding.
Pros: Large capacity, doesn’t affect handling too much
Cons: Heavy, expensive, needs to be locked
There are a number of smaller bag options such as saddlebags and handlebar bags. These are designed for smaller items such as puncture repair kit, phone, wallet etc.