Hive Tool - These come in many different forms and sizes. This is the beekeeper's all purpose tool. Mostly used to assist in separating the hive boxes and other propolized components, and removing frames and burr comb.
Bee Brush - Used to brush bees off of frames when harvesting honey. Note that brushing bees, even gently, can excite them to sting. Many beekeepers use other methods to drive bees out of a honey super before harvesting.
Frame Cleaner - Used to clean the groves and crevasses of frames.
Frame Grip - These come in many different models. A good frame grip will have a return spring, and will securely hold the frame to reduce the chance of dropping the frame. Some frame grips come with a small hive tool built in, but this built in hive tool is of limited use at best.
Smoker - These come in many different models and sizes. Smokers are used to distract the bees while the beekeeper inspects the hive. A good smoker should have a guard around it to avoid burning the beekeeper, and good airflow to keep the flame from dying. Contrary to what the Bee Movie depicts, smokers do not sedate honeybees.
Veil - The veil is a must have for any beekeeper. They are used to protect the beekeeper from stings to the face. A good veil should be easy to secure the bottom of the netting without leaving a space for bees to get in. There are many options of veils, including some that are built into a full bee suit.