An appointment type is a service that clients can schedule with you. They are created as stand-alone items that clients book and pay for one at a time. These are great if your clients will want to book your services on an ad hoc basis, or if you offer are "one time" services. These are the basic building blocks in Acuity, so you have to have appointment types for your clients to book!
Appointment packages are a set number of appointments for a fixed price. Once clients purchase a package they'll get a code they can use to redeem for appointments. Their appointment balance is automatically tracked. Appointment packages are great if, for example, you want to offer a discount to clients who book a block of appointments at once! You can also create appointment packages that act as gift certificates if you want people to be able to buy your services as a gift for a friend or loved one!