Basically, if someone offers a connection, it means that you can link the family line with theirs.
For example:
If I had a character (called Bob Robinson), I could connect them with another family line (let's say Julia Noels).
they could be cousins or step-brother & sister, etc.
Another example:
If someone is willing, you can be in the same line as their family.
Your char could be Bob Robinson, and someone could offer you Beth Robinson and they are 'connected' as family.
does that make sense?
Connections themselves are just the way that your character is linked to another character.