Electronic – arduino – Connecting components to an arduino without a breadboard

What are those "larger components"? The only larger thing is the relay, and most relays will fit on a breadboard.

This is how you control the relay (the coil is shown next to the diode), it assumes you can connect the 12V's ground to the Arduino's. Resistor, transistor and diode are normal, small components. This relay is just a few cm long, wide and high. It can switch 10A and 230V. If you tell us more about what you want to switch I can give you more directed advice.

edit re your shopping
The relay requires 90mA from your 5V power supply. That will add a couple of hundreds of mW in the Arduino's voltage regulator. At 12V in that would be 630mW, which is a pity. If you have 12V in it would have been better to use that for a 12V relay.

The TIP31 transistor is overkill. It's a power transistor, and they don't have very high \$H_{FE}\$ (the current gain). Next time go for a TO-92 general purpose transistor like the BC547. The BC547B variant has an \$H_{FE}\$ of minimum 200. Go for a high \$H_{FE}\$. This one is still OK at an \$H_{FE}\$ of 100, but I would take a safety factor, and calculate with 40. Then the base current has to be 90mA/40 = 2.25mA. A 1k\$\Omega\$ base resistor will give you 4.3mA, so that's OK.

I think your main problem is that you aren't assigning a variable or using an if statement with your digitalRead(PIR_override);, but even with that you may miss the button press in the loop. If it's a momentary switch then it goes high/low for only the time that it's pressed, so the processor would have to see this at the right time in the while loop. You could just hold the button down for about a second or you could connect the button to pin 2 or 3 and use an interrupt, which it looks like your button is connected to pin 3 according to your code. I think this is a better solution. In the embedded world interrupts are your friend.

So, according to the documentation (https://www.arduino.cc/en/Reference/AttachInterrupt), you could do something like this:

const byte PIR_override = 3;        // the number of the switch pin
volatile boolean override = false;          // Use this to know if your currently  in override mode or not when looking at the button press

Inside Setup() add:

pinMode(PIR_override, INPUT);
attachInterrupt(digitalPinToInterrupt(PIR_override), motionOverride, LOW); // the sense may need to be changed - from your image it looks like the button goes low when pressed

So now motionOverride will be called when the button is pressed. Just put what you want to happen inside this function. Something like this:

void motionOverride()
{
    if(override) // overriding motion and the button has been pressed - got to PIR mode
    {
        digitalWrite(overridestatus, LOW);  // turn override status LED OFF
        digitalWrite(relaypin, LOW);  // turn relay OFF - you may not want to do this... depends on what you want
        override = false;               // set the override flag
    }
    else // not overriding motion and the button has been pressed - go to override mode
    {
        digitalWrite(overridestatus, HIGH);  // turn override status LED ON
        digitalWrite(relaypin, HIGH);  // turn relay ON
        override = true;                // set the override flag
    }

   // I guess Arduino takes care of clearing the interrupt for you
}

Then inside your main loop() just look at the override flag (Note: I did not check any of your PIR motion detection logic, and hopefully I didn't mess up your brackets there):

void loop()
{
     if(!override)
     {
         val = digitalRead(inputPin);  // read PIR input value
         if (val == HIGH)  // check if the input is HIGH
         {           
              digitalWrite(relaypin, HIGH);  // turn LED ON
              if (pirState == LOW) 
              {
                   // we have just turned on
                   Serial.println("Motion detected!");
                   // We only want to print on the output change, not state
                   pirState = HIGH;
                   delay(DELVAR);    // maximum delay is 32776 millisecons,
                   //  delay(DELVAR);    // so add multiple delays together to get
                   // delay(DELVAR);    // so add multiple delays together to get
             }
        } 
        else
        {
            digitalWrite(relaypin, LOW); // turn LED OFF
            if (pirState == HIGH)
            {
                 // we have just turned of
                 Serial.println("Motion ended!");
                 // We only want to print on the output change, not state
                 pirState = LOW;
            } 
        }
     }
}

Try that out if you want. If there's a mistake I'll try and help with it.