on/off any appliances

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Each time the IR receiver detects an IR pulse it turns on Q1 briefly. When Q1 turns off it clocks the counter (pin 14). If the count is 1 (pin 2 high) Q2 switches on and energises the relay coil and LED D3 turns on. If the count is 2 pin 4 goes high and so does pin 15 so the counter resets to 0 immediately. This means pin 3 goes high, pin 2 goes low, the relay is de-energised and LED D2 lights.
 
Your schematic has no part numbers so are we supposed to guess which ones they are?? There are millions of parts like those.
The output of some (but not all) IR receiver ICs goes high when it receives modulated IR pulses which turns off the transistor Q1.

The transistor drives the BOX (IC1) that has no part number. The datasheet for the BOX says what it does and shows how to use it. It is probably a clocked flip-flop divide by 2 circuit.

Since the circuit is powered from 5V then the BOX is probably an ancient TTL logic IC.
 
the BOX is probably an ancient TTL logic IC.
Certainly a bit of a puzzle requiring detective work. The OP calls it a counter and the pin numbering is consistent with a CD4017. We don't know if the IR Rx goes high or low on receipt of a pulse: but either way, the positive-going (leading or lagging) edge can hopefully clock the counter. I have some concerns, though, that the clock edge will be slugged by the capacitor on the Rx output.
 
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