Why were my contacts synced to my mother's iPhone?
Sharing a single Apple Id can save you money when downloading apps or music, but can cause confusion when attempting to use other Apple services associated with your Apple Id. In this particular instance I would venture your contact showed up on your mom's phone because she signed into iCloud with your Apple Id. By default iCloud syncs a lot of information:


Contacts
Calendars
Reminders
Safari
Passbook
Photo Stream
Documents and Data
Find My iPhone


If you go to Settings > iCloud you can toggle these services to on or off. Unfortunately, what I don't know is if toggling contacts off will remove the already synced contacts.

About a year ago I asked the question: how many Apple IDs should one household have? This spawned a lot of great answers and I definitely recommend reading it. To summarize it's okay to use the one account for purchases, but I would recommend having additional accounts for iCloud, Facetime etc.