Importance of laser numerical aperture in selecting photo diode
First a laser is one thing, and an LED is something else. If you bought something marketed as a "LASER LED" that's just a marketing or trade name and it has no technical meaning. 

Numerical aperture is the sine of the half-angle of divergence of a beam. An N.A. of 0 would indicate a perfectly collimated beam (a physical impossibility) while an N.A. of 1 would indicate the beam spreads over a full \$2\pi\$ steradians, i.e. it emits into the entire half sphere in front of the emitter.

If you're talking about simply directing the beam from an emitter toward a detector, a lower NA would mean you can get a larger fraction of the total light onto any given emitter at some distance away. A larger NA would mean you don't need to be so accurate in pointing your emitter in exactly the right direction. 

More likely if you're trying to cover more than a few centimeters of distance, you'll need to use some kind of optics. Since this would be designed to take light from a point source and project it in a collimated beam, optics engineers call it a telescope. The NA of the emitter gives a requirement on the design of the telescope if you want to capture most of the light from the emitter. 

The N.A. of the emitter has very little impact on the choice of detector unless the distance between them is very short (a few mm at most) and you want to choose a detector diameter to capture the whole emitter output.