This is an impossible question to answer. A lot of things affect the snare sound. I'd say the two biggest factors in snare sound are shell type and heads. I say shell type rather than shell material because - the size of the drum, the material the drum is made of, and the venting all affect the sound. A metal shell is thought to be brighter and have more of a "crack" than wood shells, which are usually warmer and darker. Even then, some metal shells are brighter/darker than others depending on the metal/alloy used, same for wood shells. Steel shells, for example, will sound much brighter than brass or copper shells, while walnut shells give a darker sound than maple shells.
The size affects the sound as well - pitch is determined by drum diameter as well as head tension, though for snares the difference is less noticeable especially when the snares are on, since most snare heads are tuned fairly tight. The depth affects the response and body of the sound - a deeper drum gives a more full-bodied sound, while a shallower drum makes the drum sound sharper and more responsive.
Beyond this, head choices and tuning are the most important factors. On a snare, the issues will most likely be - single ply or double ply? Power dot? Edge ring to control overtones, as well as anything else used to control overtones (moongel, tape, tissue, etc)? Vented head (drier sound)? Most drummers use coated heads for snares, so that tends not to be an issue. Also consider the weight of the snare-side head - a heavier/thicker head will result in a slower snare response. The tuning, naturally, also affects the sound of the snare.
After those two factors you still have issues like the choice of snares, whether the snares are tight against the head or loose (having the snares too tight against the head will make it choke, too loose will lessen the snare response a lot), stick choice (particularly the tip shape - oval tips darkest sound, round tips brightest sound - this difference is quite minimal though, I think), choice of hoops (flanged, die-cast or wood - flanged hoops give a more open sound, die-cast gives a more focused sound, wood hoops give a warmer, woodier sound) and so on. The choice of hoops and drum shell will also affect your cross-stick and rimshot sound.
All this work to get the live sound right - now when you have to mic the drum under studio conditions, you have the issue of whether to mic from the top or bottom or both, and then how to mix the two later if you use both. Not just that, the type of mic also affects the sound. Also, close mic'ing means the tuning will be different from if the drums are being played unmic'ed. When drums are unmic'ed, they need to be tuned open with as little muffling as possible in order to project; when mic'ed, this resonance has to be cut down significantly but not so much as to remove all the tone.
So you see, this is an impossible question to answer - I hope I've given you more insight on the factors that affect the snare sound, although if you're actually looking for how to achieve that sound, it will take a lot of trial and error to find a snare sound you like. Hope this helps.
