Generically, The easiest way to do this is to set the Launchpad to "User 1" mode, in which each square button transmits one MIDI note at a fixed velocity (100, if I remember correctly). Most any sampler can play individual drum samples in response to MIDI notes.
But if you're specifically using Ableton Live, and I presume you are since the Launchpad is advertised as a companion for it, you can do this very cleanly using a Drum Rack device. The User 1 MIDI layout maps the lower-left side of the Launchpad to the MIDI notes used by the default view of the Drum Rack, and the notes ascend through the left half of the Launchpad, then the right. This diagram explains this using note numbers instead of note names, but should get the point across:
This is very convenient for Drum Racks, which group pads into four columns:
So, to set all of this up, make a new MIDI track, drop a Drum Rack device into it, set the MIDI input to your Launchpad, and start dragging samples into Drum Rack slots. Set the Launchpad to User 1, and start playing. You can safely switch the Launchpad over to Session or Mixer mode temporarily if you want to control Session View or Live's mixer controls; you don't have to reconfigure your Drum Rack track.
And, just for fun, if you create another MIDI track, set its input to your Drum Rack track (so you're duplicating the MIDI messages that get sent to the drum rack), you can set that track's output to the Launchpad, which will light up the LED pads in response to your playing, or to any MIDI clip data - whatever goes into your Drum Rack. Just be sure to disable this if you switch to Mixer or Session modes, as the LED feedback doesn't make any sense in those modes!
Put all of this together, and you can trigger drums and samples and clips like the guy in the promo video: