The chip on the left is a PCF8574 I2C expander to allow the pi to drive the HD44780 LCD screen using only 2 data lines.
The chip on the right is a MCP3008 8 channel analog to digital converter connected to the SPI interface on the pi.
The boiler status is derived from an LDR attached to the pilot light window on the front of the boiler. It was originally connected to a transistor circuit to convert the LDR resistance to an on-off digital state. But calibration was easier to do in software after connecting the LDR to the MCP3008 and getting a value between 0 and 1023.