How do Pokemon that require friendship evolve?

All friendship evolving Pokemon require 220 friendship points in order to evolve. Friendship can be gained by various means as outlined by these charts. Generally you want to walk around with it a bunch, level it up, get the Pokemon in question massages and the like, and give it the Soothe Bell. Don't feed the Pokemon herbal medicine and don't let it faint.

Note, in this answer, friendship and happiness are synonymous.

For such Pokemon to evolve, yes, you need to match a certain number before you can evolve them, but this number is not really visible, kind of like how EVs were before the introduction of the Effort-o-Meter, where even so, you only have a graphical representation without actual numbers (though you can work the numbers out if you keep track of the points you get in each stat).

While the maximum possible happiness points is 255, you need at least 220 happiness points for a Pokemon that requires happiness to evolve. Once it reaches the requirements, it will have to gain at least 1 level to evolve.

To gain happiness, there are several ways to do so (most figures that follow are from B2W2 games and assumed to have been kept the same):

• Walk many steps (you get 1 to 2 points for every 128 steps you make while the Pokemon is in your party)
• Giving vitamins to the Pokemon (you get 2 to 6 points per vitamin)
• Leveling up the Pokemon (you get 2 to 6 points per level)
• Using the soothing bag (20 points)
• Using stat-reducing berries (2 to 11 points per berry)
• Giving them a beverage from the Juice Shoppe

Having the Pokemon in a Luxury ball and/or making them hold a Soothe Bell further adds to the points you usually get (except those you get from the Soothe Bag), and lastly, you get bonus points if your Pokemon gets Happiness points at the place you met it.

You can check the happiness of a particular Pokemon by talking to a girl in Laverre City (picture from Serebii.net):

Statement                                                           |  Points
--------------------------------------------------------------------+---------
"It's amazingly friendly toward you! It must be so happy            |  250-255
 spending every day with you!"                                      |
--------------------------------------------------------------------+---------
"You must really like your <Pokémon> and always keep it by your     |  200-249
 side!"                                                             |
--------------------------------------------------------------------+---------
"Well, I think you and <Pokémon> will be an even greater combo      |  150-199
 someday!"                                                          |
--------------------------------------------------------------------+---------
"It's a little bit friendly to you… Something like that."           |  100-149
--------------------------------------------------------------------+---------
"Hmm…I think you have a lot of time ahead of you to get to know one |    50-89
 another better."                                                   |
--------------------------------------------------------------------+---------
"Don't tell me! Are you just letting it get knocked out in Pokémon  |     0-49
 battles?!"                                                         |

Further details on Bulbapedia and Serebii.