What is the effects on depth of field, signal to noise ratio and image resolution when reducing the aperture?
Aperture is measured as a fraction, and the number given is the denominator, so we get the paradoxical situation that a "smaller" aperture means a bigger number and vice versa (e.g., for a typical camera lens the largest aperture might be f/2.8 and the smallest f/22).
Depth of field is easy to answer (and has been many times): a smaller aperture increases depth of field (and a larger aperture decreases depth of field).
The aperture has no direct effect on signal to noise ratio. Depending on how you shoot, what your camera automates, etc., a smaller aperture might indirectly lead to a change in something like the camera's ISO setting, which might (in turn) affect signal to noise, but the aperture by itself doesn't.
In most typical lenses, there two different sources of un-sharpness. Any lens has aberrations, which reduce sharpness. At least in most typical cases, these tend to get worse (reduce sharpness more) at larger apertures.
At the same time, the aperture itself causes diffraction. This works in the opposite direction -- a smaller aperture increases diffraction, limiting the sharpness possible at any given aperture.
Between the two, a typical lens will have resolution figures that are at their highest somewhere around a stop or two from the widest aperture the lens supports1. As you open the aperture further than that, the aberrations reduce your resolution. As you close the aperture further, diffraction reduces the resolution. That, however, is only considering the sharpness at the plane of best focus -- when you're taking pictures of 3D objects, increased depth of field can make the picture look sharper overall, even if if reduces sharpness at the one depth where the sharpness is truly at its best.
Although it isn't technically part of resolution, I should also add that many lenses display "veiling" (aka "veiling glare", etc.) at or close to their maximum aperture. This can (sometimes dramatically) reduce contrast. A great deal of what most people perceive as "sharp" pictures is really based on contrast, so this can reduce the appearance of sharpness even when the resolution is actually quite high.
1 I should add that that's only a general rule of thumb though -- a few lenses show their highest resolution with the aperture wide open. Others need to be stopped down even more than two stops to get to their maximum. Really fast lenses (e.g., f/1.2, f/1.4) often need to be stopped down more than two stops to get to their maximum, while really slow lenses (f/5.6, f/6.3) are often closer to their maximum resolution wide open.