I am using a 2 × 16 LCD. I have connected this LCD to the Arduino UNO without any potentiometer. The thing is it works fine but when I attach any sensor or module it starts showing weird characters. Even when I upload a simple 'Hello World' program to display on the LCD, when a sensor or module is attached, it shows weird characters....
Post your complete sketch (program code)! If you don't you waste time while people ask you to do that.
When you post your code put it between [ code ] ... [ /code ] tags. You can do that by hitting the </> button above the posting area.
If you get an error, post the error (copy and paste). Not just "I got an error". If the error is shown on the display, post a picture of the weired characters on the display.
Link to the datasheet of the display you are using. Point the helpers to the page where you found the information, that you can omit the potentiometer.
Post YOUR circuit - not just any circuit you have found on instructables.
Show real pictures where helpers can identify each module and each single connection.
Your diagram - such as it is - shows pin 3 of the display connected to pin 6 of the UNO. That is essentially useless and in any case you have not shown you code.
You do not really need a potentiometer, what you have now is equivalent to connecting pin 3 to ground so you might as well do that and it will probably work better.
You can try a resistor of up to 1k in series with the connection to ground to see if you like the contrast setting better.