You hand is always the best five card hand you can make, so in this case:
Your hand is : Pair of 10's with an Ace, Queen, and 4
Their hand is: Pair of 10's with an Ace, King , and 4
The King beats the Queen, so their hand is best; they win the pot. The 4 is also relevant since it is part of the best 5 card hand you can make.
A tie occurs only if your best five card hands are the same. Suppose the board was:
10 A K Q 2
Then if you had 10 9, and your opponent had 10 3, you would tie since you would both have:
Pair of 10's with an Ace, King, and Queen
The 9 you have is irrelevant since it isn't part of your best 5 card hand.
If you always think "best 5 cards" it becomes easy to tell when to apply kickers and when not too. For example a full house is a five card hand, so having a full house with an Ace kicker is meaningless. If two players have the same full house, then they tie (assuming no better hands).