Fallout Shelter

Where do babies come from? Storks, of course!

In order to, erm, grow the population from the, um, inside out, you need to place a male and a female into living quarters. Charisma has a massive effect on this, particularly the time it takes for the two to decide to get together. First they talk, then they dance, then they run into the back of the room... he runs out happy, she limps out, holding her back and wearing a yellow shirt. She is now pregnant.

You'll need to upgrade your living quarters, now.

A key to making this faster is to have both male and female max out their charisma. Have each wear clothing that increases charisma, and it'll go faster.

Pregnant women still use any gear you give them, so any clothing perks they have will still apply. This is important in maximizing efficiency. However, you'll want to consider taking good weapons away from them while they're pregnant. Pregnant women run away from fights.

After several hours, you'll be told that a baby is coming. Click on the little green baby head icon, and the baby will then join the population, and she will revert to whatever gear she is wearing. If you are working to grow the population quickly, you can again place her in quarters with a man, and the circle of life repeats.

Of note: babies will not be born when your vault is full (they stay pregnant), so either dwellers will need to die or be exiled first.

This achievement requires you to birth 25 kids. To do so you need to get you female dwellers pregnant by sending them to the living quarters along with at least one male companion. It is preferred that both the female and male dwellers have a decent charisma stat to speed up the process of getting the female pregnant. They'll say some funny/cheesy pick up lines, dance a little, then head off to do their thing in privacy. When you reach that point your good ol' buddy vault boy will show up in the bottom right of the screen with a very 'nice bro!' thumbs up to let you know they are on they're way to bang. When they are all tuckered out and have had enough fun for the night they will emerge and the woman will now be sporting a yellow shirt and an inflated stomach no matter what she was wearing before. Now just wait 3 real life hours and you will see a nice overly saturated neon green icon in the shape of a baby with a pacifier in its mouth above said prego womans head. Press or click on the icon and you will have birthed one child (I thought it was supposed to be long and painful?). To this 24 more times and the achievement is yours!

I have a guide explaining this process if you need just a tad bit more detail:

Put a male dweller and a female dweller in any LIVING QUARTERS and let them hang out together for a bit. Eventually they will begin to dance to each other and say nice things. After a while they will go in the back and smiling emotes will begin to float around. Moments later they emerge and the woman will be wearing a yellow shirt and be pregnant. Wait 3 hours* and youll have a baby. Do this 25 times and youll unlock this achievement

Things to note about pregnant dwellers and kids:
You can put a pregnant woman back to work but not out to the wasteland. In an emergency she will run away, she will not fight while pregnant. So outfits and weapons are worthless on pregnant dwellers. You cannot make babies with dwellers from the same family...that would just be creepy anyway. You can make babies repeatedly with the same two dwellers. The woman will give birth to a dweller child, which will add to your population. They too are worthless in a fight. They cannot be equipped with gear until they mature to adults after 3 hours. When born they have a mixture of the parents stats. Higher stats on parent means higher stats for the children. Higher CHARISMA on the parent will speed up the time it takes before a dweller becomes pregnant

*thanks to BOngAAA for the 3 hour info