A significant other is, in the most literal form of the phrase, a person who is significant to an individual. This can include close friends and family members who have a large impact on the individual's life. Most commonly, however, the phrase "significant other" is used to refer specifically to a boyfriend or girlfriend, a husband or wife, life partner, or other person in a romantic or otherwise intimate relationship with the individual.
Due to the fact that the term "significant other" does not differentiate between gender and relationship status, such as married versus simply dating, it is frequently used in reference to an individual's romantic partner when the speaker is not familiar with the partner. For instance, if someone asks a casual co-worker to an event, he or she might say, "feel free to bring along your significant other if you would like." This leaves room for the person to understand that his or her romantic interest is invited to the event but it does not make an assumption about whether he or she is married, dating, or seeing someone of the same sex. Sticking to the term "significant other" is socially "safe" because it prevents the individual from being offended at the speaker's choice of words or having to correct the speaker and explain his or her relationship.
Following the same example, even if the person inviting co-workers to an event is familiar with each co-worker's spouse, he or she can choose to send around an email without having to individually tailor the invitations to each person's situation. It is much easier to write "feel free to bring your significant other" in a mass email than it would be to change each invitation to say "wife," "girlfriend," and so on depending on the message's recipient. This phrasing also leaves room for those who might not be dating anyone to bring along a close friend instead without feeling socially awkward.
Less commonly, "significant other" can also mean any person who is very important to the individual. For example, this usage might be found when filling out a form that asks for the telephone number of a significant other so they may be contacted in case of an emergency. In this situation, it is just as appropriate to put the name and phone number of a parent or close friend as it would be to leave a spouse's name and contact information.