I have been living and working in the U.S. for about six years, after entering illegally. I am married to another undocumented person, and we have two daughters, born in the United States. The problem is our neighbor. She doesn't like us, and keeps threatening to call U.S. immigration agents and have us deported. If she does call them, what will happen to us?
You are indeed at risk that your neighbor contacts U.S. immigration authorities (specifically, Immigration and Customs Enforcement, or ICE). However, nothing will happen immediately or automatically.
First, there is the question of whether ICE will act on this tip. They do not have the resources to follow up on every tip they get. They might simply ignore it.
ICE agents are expected to follow a policy in which they take a closer look at each individual case and decide whether to exercise something called "Prosecutorial Discretion." This means that they examine the person's or family's situation—their history of responsible work and family life in the U.S., and family ties to U.S. citizens (such as your daughters)—and sometimes decide not to initiate removal (deportation) proceedings against them.
The idea was that they are supposed to direct government resources at people who have committed crimes or present security risks or are otherwise negative forces in U.S. society. Even if ICE has already set court proceedings in motion, they can use Prosecutorial Discretion to close them; and in some cases even allow the person to obtain a work permit.
That takes us back to the question of whether ICE has the resources to act on any tip. It still doesn't. But as you no doubt realize, it's a game of chance at this point.
If ICE does attempt to remove you, agents may arrest you and/or your wife.
After an arrest, you will most likely be charged with being deportable, released on bond, and then told to appear in Immigration Court on a certain day. A document called the Notice to Appear or NTA will describe the charges against you (that you're in the U.S. unlawfully, most likely) and give you a date for your first court appearance, called a Master Calendar hearing.
If you have any defenses to deportation, you can then ask for a full court hearing at a later date (called a "Merits Hearing.") See Possible Defenses to Deportation of an Undocumented Alien for more on the possibilities, which include things like a fear of persecution in your home country giving rise to an asylum claim, or marriage to a U.S. citizen.
This agency's history of making sure that children are cared for in such a situation is not great. Make sure your children know where to go and who to contact if you are not at home when expected.
It might be wise to consult an immigration attorney in advance about your situation, so that you have already lined up someone who can start acting on your behalf and figure out where you are being held. Also see Who Will Care for My Child If ICE Arrests or Deports Me?.