What is this technique of playing music instrument called?

It appears to be a combination of a couple of different guitar techniques. The "slap", comes predominantly from Latin styled music, where the body of the guitar is slapped at the same time as either plucking or including the strings to create a percussive rhythm.

When it's being done on the frets, i.e. hitting the string hard enough to make a tone without plucking the strings it's called a "hammer on". Usually it's done with one finger, but it can be done with two or or more. Michael Hedges was a GREAT utilizer of this method, as can be demonstrated with a two finger hammer on in this video, from a live performance, around the 3:00-3:50 mark is some excellent close ups of it. (As a side note, that Live at Wolf Park was an awesome concert, and the Windham Hill CD from that year is well worth hunting down).

It is also referenced in a Guitar Player online article on slap harmonics. So you can call it slap harmonics, or multi finger hammer on, but it's a valid, recognized guitar technique.

If you are referring to this:

I'm not sure if there is a name for that kind of technique, but it exists. It contains some tapping with the one hand that actually plays the notes and the other one slaps the guitar.It is used in a way to make the guitar more percussive. A lot of people use it when they play alone and/or with other people, but there are no drums or any percussion on the group

Here is another example:

It is not 100% the same thing as the movie, but it is enough to get the point

I haven't seen August Rush but the scene in Shevliaskovic's post makes me think that it's percussive fingerstyle. It's shown only for short time though.

You can see a wide area of fingerstyle techniques (such as "hammer-on" as mentined by JohnP or "pull-off") in works of Preston Reed. Sometimes he uses many of them in the same session.