What are these controls attached to my boiler for?
This is a thermostat (or aquastat) that regulates the temperature in your boiler. Appears to be an IMIT TLSC model. Here's a similar product on amazon.

According to the instructions below, that button is a manual overheat reset. This thermostat has redundant controls - a dial you can turn (set 0-90C), but also a safety shutoff, designed to turn off at a temp just above the dial's maximum (90-110C according the instruction sheet below). That way if the dial-operated controls get stuck on somehow, there is an extra check to avoid overheating your boiler. (Overheating the boiler is bad because it creates pressure, leading to leaks or explosion.)



The button is usually covered by a removable plastic cover, which appears to have been removed here.

It's possible you also have a reset control located elsewhere on the boiler, e.g. on your oil burner. A reset control on the burner would cut off fuel to prevent overheating. The pictured thermostat's control would cut off the electrical signal to run the burner. These systems are not entirely redundant - you're protected against a wider range of failures by having both.