A courier/messenger will run into you and hand you adoption and land letters. This part of the Hearthfire Strategy Guide covers adoption. If you have not run into the courier, just walk around town for a few minutes, sometimes more. Try waiting several in game hours (using the wait menu) and keep wandering around town. Eventually, the courier should show up.

The courier tells you that there is a new lady in charge of the Orphanage in Rifen, and she asked him to hand out notices to everyone he met.

He gives you an item. Open your inventory. Under "Books", search for a letter labeled "Consider Adoption". The letter seeks a good home for the orphans of Skyrim. It calls out to provide a stable home for innocent children left without parents, the true victims of the war in Skyrim. You can make a difference in these kids' lives if you can adopt them, rescuing them from a life of poverty, hard labor, and misery. It tells you to inquire with Constance Michel from the Honorhall Orphanage in Riften for more info.

You receive the following objective:
Speak with Constance Michel about Adoption

Fast Travel Riften and locate the Honorhall Orphanage.

NOTE: While the Riften Orphanage has many of the kids that you can adopt, there are more children who could be potentially adopted in other areas of Skyrim. For a complete list of boys and girls that you can adopt in Skyrim Hearthfire, see this wiki:
Skyrim Hearthfire: Adoptable Children

Enter the Orphanage and talk to Constance Michel. She tells you that this war has created so many orphans, and so few are wiling to open their hearts to them. Ask her if you could adopt one of her children. She tells you she will need to ask you a few questions.

Answers to the Adoption Questions

Question: "Your name?"
Answer: Choose your character's name

Question: "What do you do for a living?"
Choices here will depend on what your character has achieved, and possible answers include those such as:
"I'm a Dragonborn"
"I'm Harbinger of the Companions"
"I'm Arch-Mage of the College of Winterhold"
"I'm a Thief"
"I'm an Assassin"
"I"m an Adventurer"
"I'm a Mercenary"

Answer: Choose an answer that would impress her, such as Dragonborn or Archmage.

Question: Where will your child be living?
Answer: This depends on whether you own a house or not, and whether you have a room for the kids or not.

If you have not met the accommodation requirements of adopting a child, she tells you that she needs to be sure your child will have a safe, stable home to grow up in. She wants you to show her that you can support a child. Buy a house, furnish it, then come back to talk to her. If you try talking to her again before the adopting requirements are met, she will tell you: "I'm sorry, but until you own a home with a room for your children, I can't allow you to adopt anyone".

Hearthfire Child Adoption Requirements

House in the City (Traditional): If you want to adopt a kid but don't feel like building a house just yet, and you have previously purchased a house in one of the cities, the courier should have handed you a letter from the Jarl's Steward offering you an additional furnishing option for your house. For example, if you own Breezehome (Whiterun house) you will get a letter called "Breezehome Furnishings" offering you to purchase a children's bedroom set for your home, updating your Home Decorating Guide accordingly. Speak with the Steward (in case of Breezehome it is Proventus Avenicci, the Steward of Whiterun) to purchase this furnishing for your house.

Purchasing the children's bedroom set for any of your existing homes will satisfy the requirements for adopting a child. Speak with Constance Michel again after you are done.

IMPORTANT: The kid's bedroom you build in this manner replaces an existing part of your home, usually the enchanter station or alchemy station. Be sure to move any of your weapons, armor or other items you have placed in that area, since a bug/glitch exists where your house gets furnished over the items and they are lost forever. So be sure to move any valuable items to a safe place before adding the child's bedroom.

Purchase a Deed and build your own house on your plot of land (NEW): This new feature in Hearthfire lets you purchase land and build a house on it. The process is covered in previous guides:
How to buy a deed and plot of land in Skyrim Hearthfire
How to build a house in Skyrim Hearthfire

The important thing to be aware of when building your house is that it needs children's beds (not just regular beds) and a dresser/chest for each of your kids to be able to move in. You can put the small dresser and child's bed in the main hall, you do not have to build a separate bedroom wing for them.

Adopting a child
Once the adoption requirements are met, talk to Constance Michel at the Honorhall Orphanage again to adopt a kid. Tell her you've built/purchased a home and furnished it, and ask if you can adopt a child now. She will agree. She tells you to go meet the children and decide who you want to adopt. Once you've made a decision, Constance Michel will collect their things and be sure they make it over to your house.

Talk to the children and decide who to adopt. You can ask each child to tell you more about himself or herself, or offer to adopt him/her from which point you will refer to him/her as son or daughter. The kid will tell you something like "just let me get my things and say goodbye to everyone. I'll meet you at home. Thanks! Thanks again!". Inform Constance as well of your decision, then go to your home. It might take a few days for the kids to arrive there and meet you.

Upon successfully adopting a boy or a girl, you will receive the achievement:

Proud Parent
Adopt a child