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: