The option to build support relationships opens up right before Chapter 2 begins.
When two characters continue to fight together or help each other, their relationships may improve. This will be shown by red hearts floating above the characters on the battle map.
Note that not all characters are compatible with one another. The "Support" menu, which can be accessed before beginning a battle, shows which characters are compatible.
Your Avatar has the perk of being able to support everybody.
Once a relationship has sufficiently developed, the two characters will be able to "Talk" in the "Support" menu, when outside of battle. These support talks give a little insight into the characters and cements the relationship with a support rank.
Support ranks start at C and then go up to B, A and finally S, with a special talk before each rank up.
Characters can have any number of support relationships with other characters, but only one S Rank support.
When characters fight adjacent to or paired up with a character whom they share a support rank with, their combat stats will increase slightly, as will their chance of performing a Dual Strike or Dual Guard. The higher the support rank, the higher the bonuses.
Each time you complete a map, check to see if any new support ranks can be created or improved. Not only will your units fight better, but you'll also learn more about them!
An S Rank support can only be formed between opposite gender characters (and if they aren't related, eg. siblings or parent/child) and it results in the two characters getting married.
Since your Avatar can support everybody, this means they can marry anybody of the opposite gender.
Upon completion of Chapter 13, depending on which characters have married, you will be able to recruit their children in optional sidequests and enlist them into your army. The children will inherit attributes from both their parents, such as stats and skills.
Note that due to story-reasons, if you do not marry Chrom with somebody, he will automatically be married at the end of Chapter 11. The person he marries will be the bachelorette whom he shares the highest support rank with. If nobody is available, he will marry a village maiden.