Family bonds are only featured on Tamagotchi Connection Version 5 (Familitchi) and Version 5.5 (Celebrity Familitchi). They determine how close the members of the family in play are to each other in percentage (i.e. 20%, 50%, 100%).
The purpose of family bonds are to try and get them as high as possible before sending your Tamagotchis off to get married. In order to raise a family such as a Mame Family, Kuchipa Family, etc. you must raise your family bonds to 100%. To see your family bonds, select the health meter and press the A button once. The screen will display the Bonds Percentage and what type of family you have (i.e. Blended Family, Meme Family, etc.).
How to Raise the Bond Percentage
Raise the family bonds by playing games or buying and using items that all the siblings in the Tamagotchi family can use together (i.e. basketball, top hat, etc.). Bonds can also be raised by answering the training call (sixth icon). The Bond percentage rises quicker after the first generation since the games played with parents increase the Bonding more than standard games.
If the C button is pressed while on the normal playing screen, it will show a cut scene of the Tamagotchi family. If the bonds are low, they will be shown looking left to right followed by question marks appearing over their heads in confusion. The next stage is when the lead tamagotchi runs up to the screen and smiles. The stage following, shows all of the Tamagotchis skipping together off the the screen and as the come back, they all trip and fall and sit up together laughing. The stage after that, the Tamagotchis will all kiss each other's cheeks and a big heart will form. The last stage is when all the Tamagotchi siblings come up to the screen.