Marrying your characters is entirely optional and you can finish the game without even marrying anyone. There are two major considerations when you're playing cupid; either you'll marry the characters based on their personalities or whoever you think is the best partner for them or you marry them with calculated consideration of their offspring's skills, stats, Re-Classing option and more. The first method is carefree and basically having fun with the game without worrying about the in-depth details. The second method is very mechanical as you have to make tedious preparations, grinding, farming, and even prepare a personal pairing chart that will help you determine who will marry who, depending on the skills they can pass down to their offspring. The second method is technically breeding the perfect offspring units.
Offspring characters are optional units conceived from specific male characters who reached Rank S support (married) to compatible female characters. The offsprings will carry the hair color of their mothers and will inherit the skills of the parents. Because of skill inheritance, offspring characters can be very powerful units especially if the parents got their advanced Classes (or better yet, re-class to another class and learned additional skills) before marrying.
Azura and the female Corrin are the only two female characters in the game that can have their own offsprings derived from them. Corrin's child, Kana, will have the opposite gender from Corrin. Male Corrin can sire two children only if married with Azura. Female Corrin can have two children whoever she decides to marry. Children can also inherit the Royalty status (ability to trigger Dragon Veins) if any one of the parents is a royalty.
Recruitment
The children can be recruited by completing objectives in their own chapters called Paralogues. Their level will depend on how far you are in the story. They'll start with base Classes but if you start the paralogues when you're around Chapter 18 in the main story, they'll come with Offspring Seals, which will allow them to get promoted to their advanced class even during the paralogue itself. Check out this guide's dedicated section for all Paralogue chapters in the game.
Marriage
Offspring characters can marry each other but they won't produce any children. The exception to this rule is the avatar who can marry any of the offspring character opposite to the avatar's gender. In that setup, the avatar's kid will have grandparents (offspring character's parents).
Skill and Class Inheritance
All offspring will start with their default class the first time you get them. If the father's default base class is different from the child, the father's base and advanced class will be added as additional re-class options to the child. If the child and the father's default class is the same, then the father's secondary class will be made available instead.
Offspring characters will have their default unique skill and will inherit at least one skill from the last slot of their parents. You can edit this through Equip Skills menu. This will dictate the skills that will surely get passed down to the child. If you're able to plan ahead and was able to pass down the skills you want the offspring to inherit, it's safe to re-class and learn new skills instead. If the offspring has already learned the advanced Classes for his/her class line, it's recommended to re-class to maximize the potential to learn new skills.
Bond Units
These are special units that are "born" through the bonds of two different avatars. This is achieved by two different players exchanging gifts to each whenever they visit each other's castle. Multiple exchanges are required and once the bond is high enough, a bond unit is born, inheriting the skills of both parents. Bond units born from highly-reclassed/ advanced avatars can potentially get powerful skills in one package. Like einherjars and captured characters, bond units have no support ranks.