Almost anyone can suddenly find themselves in the role of a "first responder" to a mental health crisis. Fortunately, there are many places you can turn if a person considering suicide asks you for help. There is nothing wrong with trying more than one of them if necessary.
Online resources
Good online resources for suicide prevention include the following:
Many others can be found with careful use of any search engine.
Local resources
A local telephone directory is often the best resource for finding help in your area. Many phone books list a local suicide prevention number in the inside of the front cover, along with other emergency numbers. The description might be "suicide prevention" or the more general phrase "mental health crisis" depending on the local area.
If no number is listed inside the front cover, check the public health or mental health services listings for each nearby city, county, or other local government.
Private charitable agencies, including many religious organizations, may also provide counseling services at low or no cost. Use caution when approaching religious groups about mental health issues. Some cling to outdated theories about the nature of mental illness, including surprising numbers who still think "demonic possession" is the underlying cause.
Notes
Suicide prevention, as discussed here, has nothing to do with "forcing" anyone to go on living if they do not wish to live. Its primary goal is to help people who want to live, and who consider suicide only because of a short term crisis situation. Other ethical issues, such as "physician assisted suicide" for the terminally ill, are comparatively rare. Most suicide prevention agencies focus on helping people deal with crisis situations.
Be careful not to wait too long when dealing with a person who does not respond to efforts to help. You could fall into a pattern called "compassion fatigue" which can lead you to respond harshly in ways you might regret later.
Remember, the simple distraction of talking with someone can sometimes be enough to help a person get through a period of crisis. This is especially true for people with various cyclic mood disorders, such as manic-depressive patterns or bipolar disorders. Don't discount your own ability to help just by listening, even if you've never had any training on how to help a person in crisis.