Once you have the rankings down, all you need to do is combine the board with the hole cards to make the best 5 card hand.

Couple examples:

You have 22 | Board is 4 5 6 7 8 | your opponent has T9

Since there is a straight on the board, the best 5 card hand is 4 5 6 7 8

Your 22 doesn't "play", meaning that it is irrelevant to the hand since a straight is better than a pair. Your opponent however, can combine his cards with the board for a hand of 6 7 8 9 T. So his straight is better than your straight, he wins.

You have QQ | Board is K K K J K | Your opponent has A2

It looks like you have a full house here. KKKQQ... but actually your QQ doesn't play here because there's a 4 of a kind on the board, which means your hand is KKKKQ. Your opponent would win this hand because he has a better kicker to the 4 of a kind. KKKKA (this would be a disgusting bad beat for you)

You have 47 | Board is 4 7 2 | Your opponent has 88

You have flopped 2 pair, right now your hand is 77442. Your opponent's hand is 88472 (one pair). So you are winning.

Next card is a K. This changes nothing. You are still winning with 2 pair.

River is a 2. You have now lost the hand.

Board is 4 7 2 K 2

Your hand is 7744K (this is called 7's up, because the higher pair is a 7)

Your opponent's hand is 8822K (he has 8's up, which is better than 7's up)

It doesn't matter what the lower pair is, only the higher pair. Which is why he wins.

If the river had been a K, the board would be 472KK

Your hand would now be KK774 (Kings up)

His hand would be KK887 (Kings up, but with a higher lower pair)

He wins again.