This is a good question as according to the World Health Organisation (WHO), close to 800,000 people die due to suicide every year, which is one person every 40 seconds. (WHO, n.d.) Calling the emergency services via 911 (or a local equivalent) is the first thing you should do when discovering someone is about to attempt suicide.
Emotional First Aid (EFA)
The first time I heard about any kind of "mental first aid" was around 2010 when some training courses on Emotional First Aid (EFA) were made available, run for 6 three and a half hour sessions over a 6 week period. Built on a foundation of systemic thinking and humanistic principles, EFA seeks to dispel the myth that mental health means mental illness; and so EFA was designed to help a child or young person experiencing emotional distress, before any professional help is sought.
Other Training
As members of the International Association for Suicide Prevention (IASP), there are also formal training courses run by Samaritans (UK and USA).
Psychological First Aid (PFA)
For an informal approach to learning about the subject, there is a guide for field workers written and provided in many languages by the World Health Organisation (WHO) through their Mental Health Gap Action Programme (mhGAP) called Psychological first aid: Guide for field workers (WHO, 2011). The guide is not specifically aimed at suicide prevention. It is aimed at providing guidance for Psychological First Aid for any possible reason, and therefore, the guidance can be followed whilst dealing with suicidal people.
Some Key Points (Among Others)
The prime responsibility you have is for your own personal safety
Helping the suicidal person should never endanger your own personal safety. If you get seriously injured whilst trying to help, you are not being helpful and it could add to the list of reasons the suicidal person may carry through with it.
Psychological first aid (PFA) is is not counselling or psychotherapy
so you don't need to be trained in any of the related mental health fields. Even if you are trained in any related mental health fields, you are not to treat it as counselling or psychotherapy.
Don’t force help on people, and don’t be intrusive or pushy. (WHO, 2011)
One of the things which some people might think they should do is to insist that they stay with the suicidal person and force them to talk. Not only is this going against their human rights, but someone who is suicidal because they feel they are not in control of their own lives can be pushed into carrying through with their suicide attempt with this approach.
Make it clear to them that even if they refuse help now, they can still access help in the future
If they refuse and ask you to leave them alone, you can say that you will stay nearby and if they change their mind, just call out and you will be there. The fact that you are there can be enough support.
Be aware of and set aside your own biases and prejudices.
You may have strong beliefs which may affect how you feel about that person once they tell you something. You need to be mindful that you can inadvertently display the fact that you don't approve of something which can be detrimental to the process.
After it is over, be sure to look after yourself.
Dealing with suicidal people can lead to Vicarious Trauma. Be aware of how you are feeling and how the incident has affected you. Seek emotional help for yourself where needed too.
If there is a strong chance you may be in the situation where you have a suicidal person you need to help, I would recommend at least reading the WHO guide for field workers, however it may be more beneficial to undertake some form of formal training in Psychological First Aid either through the Samaritans or a similar organisation.
References
WHO. (n.d.) Mental Health - Suicide Data.
Retrievable from: http://www.who.int/mental_health/prevention/suicide/suicideprevent/en/
WHO. (2011). Psychological first aid: Guide for field workers. Retrievable from: http://www.who.int/mental_health/publications/guide_field_workers/en/