Sam Foster (McGregor) is a psychiatrist (since his focus is on therapy rather than medications, he seems to be doing more psychologist work than psychiatrist work, though psychiatrists can and do sometimes provide therapy) who's racing the clock and his own failing grip on reality to keep Henry Latham (Gosling) from committing suicide. Since Sam's girlfriend, Lila Culpepper (Watts) once made a suicide attempt, Sam has a lot of emotional investment in stopping Henry.
**There are spoilers 
for this film in the discussion below.**
Obviously, in a story like this, where the events are viewed through the eyes of the characters, it's possible that the film creators intended for those characters to misconstrue some psychological or psychiatric details. There were several things that bothered me as I watched the film, like the strange transitions, but when I reached the end and it all came together, I was impressed. I walked around for days telling people about it.
if for any reason they can't provide services for a while. Henry tells Sam up front that he intends to kill himself in three days. Sam's initial interactions with Henry are really great. He doesn't pretend to be superior, he answers reasonable questions, and he works with Henry rather than trying to stick to some stiff script.
For example (script source) :
HENRY: So you're the substitute shrink?
Sam smiles.
SAM: I guess you could call me that. Does it make you uncomfortable that I'm filling in for Dr. Levy?
Henry thinks about it, coiling the string tightly.
HENRY: Is she your girlfriend?
SAM: No. We were classmates, actually. So listen, I read your file--
HENRY: Oh, really? May I read your file?
SAM: Fair enough. How about you ask me a question for every question I ask you?
When Sam hears that Henry is suicidal, he tenses up, saying something like, "There are certain actions I need to take if you tell me you're going to kill yourself."
Therapists have to do an assessment of dangerousness before calling the hospital—partly because it can be difficult to get someone admitted to the hospital, and the hospital (and especially the insurance company!) will insist you're sure the person is of immediate risk to himself or someone else before you send him over. Also, calling someone without being really sure there's an emergency violates the ethics of confidentiality.
Phone calls are made only after the following steps are taken, and only if the client feels s/he will be unable to stay safe. The protocol is to ask:
Even if all of the above criteria are met, options are discussed, and if possible a Safety Plan is put in place; that is, the client promises (often in writing) that he will not harm himself, and if he feels she can't hold to that, he has a series of things to do, in the order he should do them. Usually safety plans include things like calling a trusted friend and going to the hospital oneself. In situations where the client doesn't feel he can be safe, immediate hospitalization is discussed.
Though it can feel stilted to include the assessment in your story if you're not quite sure how one would go in real life, the assessment is usually where the most important (and interesting) information is revealed. Knowing that your character not only plans to shoot himself but also has a gun in the car and intends to drive out to a field after the session and shoot himself, for example, forces the therapist to take immediate action.
Since Henry has a plan, the means, and a time set three days hence, it would be interesting to ask whether there's any possibility he will do it sooner, and what would make that happen; as well as why the time is set when it is.
When Schlegel hears that Henry wants to kill himself, he says, "Tell him to take a number...but we can only hold him here for forty-eight hours [during which time we'll just] hold his hand, feed him some pills." Sam tells Schlegel that Henry has a date and time picked out, and Schlegel says, "What, he's got an appointment to off himself?" and "As you just saw, [being hospitalized] is not a real fun process. If there's any way to avoid it, avoid it."
Most people who have to be hospitalized for psychological problems say the experience was a good one, though most don't want to go back anytime soon. Again, it's like being relieved to have the broken bone set and some pain medicine to relieve the suffering, but not being in a rush to break anything else!
And while everyone uses jargon differently, most people say involuntary commitment rather than involuntary committal. (I was trained in a university-affiliated, APA accredited School of Professional Psychology, and once in a while someone would refer to the program as "soap" (SOPP) rather than S-O-P-P like everyone else did. It got some funny looks, but usually the person just liked saying "soap" better.)