Your boyfriend or girlfriend is someone with whom you are having a romantic or sexual relationship. Two people in a serious relationship are often referred to as a couple.
Have you met Tom, my partner?Lover can be used about people in sexual relationships, but it is more common to use girlfriend/boyfriend/partner in everyday conversation.
If a couple are engaged, they have formally agreed to marry. The person to whom you are engaged is your fiancé (male) or your fiancée (female).
A single parent/mother/father (single mum/dad informal), or a lone parent is someone bringing up a child alone.