What type of breakers is this and how should they be switched back on?
These are tandem/cheater breakers. They are double circuit but on the same leg/phase of the panel. When two are next to each other as shown in your picture and with the handle ties, they are protecting a 240 Volt circuit. As such, they have to be operated at the same time. To turn on or off, put your fingers on the two tabs/ handles that are connected and push in the direction to turn on or off. That would be the two in the middle with the round handle ties and the two outside with the rectangular handle ties. The only concern is that the two controlling a 240 Volt circuit are both on or off and the handle ties ensure that.