This is probably the hardest thing you will have to do as a new Muslim.  For many people, it poses the prospect of opening up old wounds, risking hurt feelings on both sides, and threatens to rip apart familial relationships.  For others, they know that they will be accepted by their parents, siblings, and other family members unconditionally.  Mash'Allah.  For teenagers, my advice would be different than advice that I would offer to an adult, especially one who is living on their own, and may already be married. Insha'Allah, I will address the concerns of young people who still live at home first.

Advice for Teenagers
Oftentimes, this is a situation which is best handled with care. There are no exact directions that I can offer to you, because how you and your family deal with this is based on a number of things: your age, your community, your relationship with your family, your previous religious experiences, your parents' commitment (or lack of) to a certain religion, and their willingness to explore new ideas.

Although it seems like a wacky idea, it has been said by other converts, and now by myself as well, that it oftentimes might be better to wait six months, a year or more to tell them. The reasons for this vary: you need to be more established in Islamic practices, and you need time to make friends and build a support system within the Muslim community. This is so that if your parents react to your announcement by attempting to "deprogram you," or schedule "an appointment" with the local minister / priest / rabbi, you will be able to rely on your knowledge of Qur'an, and the strength that being a practicing Muslim has given you. Allowing yourself time to build a support system within the Muslim community is important so that you will have friends to help and guide you, to help answer any questions or concerns your family might have, and to help you out should your parents decide that you can no longer live in their house.

If you are fearful that your family may react with physical abuse, or a kidnapping and "deprogramming" attempt (yes, it happens), please make sure that you have someone there as a witness and support. Whether you are Muslim or not, you have the right not to be abused. If your family is abusive towards you, seek the necessary help to get out of that situation as soon as possible.  

Another reason that it might be wise to wait awhile is to allow your parents to see the positive changes that Islam will bring about in you: greater care to hygiene and appearance, greater discipline in your daily activities and your schoolwork, the fact that you are not falling under negative peer pressure to drink or drug or have sex, that you are more willing to honor your parents by helping around the house, that you are more attentive in your job (if you have one), etc. Allow them time to be pleased with these positive changes, so that they may see that Islam is for the better, not just for you, but for all people. If they see that Islam is "good for you," they may react more positively when you talk with them about it.

For adults
As an adult, especially one who lives on their own, and who may be married, your parents and family are already aware that you are entitled to your own decisions. There are some converts who are not bothered one way or the other with the way their family may react because of this reality. However, for many others, it is important to them that their family respect and accept their decision. It may be difficult, especially if there are children or a disliked son-in-law involved.

An adult who's chosen Islam has to make some of the same considerations as the teen who's accepted Islam: What is your relationship with your family? What is their religious commitment, or lack of one? What degree of commitment did you have to any prior religions? How open is your family to new ideas? For the adult, some of the considerations may also include: How do your parents feel about your husband? Do your parents have a history of making you feel obliged to them for favors they have done for you since you left their house? How close are your parents to your children, if any?

Since you don't live with your parents, it will be easier to allow them the space and time that they need to deal with your announcement. Make sure that you emphasize that this hasn't changed you in any radical way, and that you strongly desire to keep your relationship with them intact. Make sure that they have access to their grandchildren, but at the same time, make it clear to them that you will not tolerate any attempts to teach them anything other than Islam, or allow them to eat haram foods or participate in haram celebrations. In some cases, it might be better if you tell them of your decision alone, so that they can't "lash out" or place the blame on your husband. Make sure that they know they must deal with you directly.

Dealing with brothers and sisters (of the biological type)
Many of us have at least one sibling, and it is important that you deal with any siblings you may have on an individual basis, if at all possible. If you are a teenager, this means talking to younger and older siblings in person, and letting them ask any questions of you that they may have. Let them know you are the same person who may argue about whose night it is to do the dishes, and that you are still their brother or sister. Stress that you still love them, especially if they are very young, and unable to understand why you don't go to Church anymore. Above all, make sure that you are acting as a proper role model for both your younger and older brothers and sisters.

If you are an adult, the chances are that you and your siblings have "issues" is great, and you may not even be on speaking terms. There is also a larger chance that you all live in different towns and states. When dealing with adult siblings, it is best to write them a letter or make a telephone call in which you can clearly explain your decision and answer any questions they may have. Prepare yourself for resentments that may pop up, especially those surrounding childhood incidents.

Don't begrudge them for their hurt feelings, and if necessary, allow them time to work through any issues that they may have: it may go deeper than your choice to become a Muslim. Assure them that you are still the same sister who loves to eat cheesecake, or watch football games.

If you are not on civil or speaking terms with a sibling, it may be best to avoid telling them your decision altogether, until you can come to a mutual understanding as family members.

For all new Muslims
The most important thing, and I can't stress this enough, is that you do not allow yourself to get dragged into a "Christianity vs. Islam," "Judaism vs. Islam," "Hinduism vs. Islam," or any sort of interfaith debate with your parents or other family members. Oftentimes, I have heard of new Muslims whose parents or siblings are in the Christian ministry, and who have been baited, taunted, and condemned by them. DO NOT ALLOW THEM TO DRAG YOU INTO A CONFLICT REGARDING RELIGION AT ALL. If a family member hurls a "judgment" at you (i.e., "You're a Satan worshipper who's going to hell!"),  do NOT respond in kind! If your relationship outside of this religious difference is salvageable, then avoid any religious discussions until everyone is willing to discuss it in a more open minded and civilized manner.

The second most important thing is that you do not allow yourself to become an active evangelizer. Avoid aggressive and continuous attempts to convert your family members, as this will only bring resentment and separation between you. The call to Islam should be a gentle call, and the best way to give da'wa to your family is to be a living example of Islam. People can get awfully stubborn when they are confronted in this manner, and they will only dig their heels in more. Do not be the cause of great tension between yourself and your family.

Finally, do not allow yourself to be baited or upset by any "anti-Islamic" things your parents and family might say. Many Americans (and Canadians) hear of Islam only from news reports and movies like 'Not Without My Daughter.' Don't allow them to mock you with jeers of "terrorist," "wife beater," and reply with slogans about "Zionists," and "hypocrites," etc. Instead, gently correct any misconceptions they may have about Islam and Muslims. If you are a woman, it is important to reassure them of your rights in Islam, and of your commitment to wear Islamic dress. If they have some very real concerns about your safety as a Muslim woman, try and arrange for them to visit the mosque and talk to the imam / amir, or to get together for coffee with other Muslim sisters.