The majority of meteorites contain some metallic iron, so the first and most fundamental test is whether the suspect meteorite is magnetic. If it isn't, it very likely is NOT a meteorite. Keep in mind, though, that there is a slim chance that it could be a non-magnetic achrondite or even a Martian meteorite (very rare). The most common type of meteorite (and the most common meteor-wrong) brought in for identification is the iron meteorite, which is solid metallic iron. Other types such as stoney-irons will have Fe-metal blebs scattered throughout, but the rock is still magnetic.
All meteoritic iron will contain some amount of nickel, so that is the first thing to look for in the SEM or microprobe. Fe-Ni metal consists of 2 types: kamacite and taenite. Kamacite is the low-Ni variety and contains about 5-10% Ni whereas taenite is Ni-rich with 30-70% Ni. You should be able to find both types in a sample, so look around with the BSE at high contrast for subtle differences in atomic number in the iron metal. Another good test is to look for Cr and Mn. Meteoritic iron contains very low levels of Cr and Mn, less than 0.02%. Since man-made Fe alloys usually contain some additives, if you find much Cr or Mn, it is definitely NOT a meteorite. If the suspect meteorite is porous or has vesicles, it likely is NOT a meteorite. And, if you find quartz, it is NOT a meteorite (although some Martian meteorites do contain silica).
You can look for unusual phases, such as schreibersite (Fe-Ni phosphide) inclusions in iron meteorites. If sulfides are present in stoney or stoney-irons, check the Fe-S ratio, since most meteoritic sulfides are troilite FeS, not FeS2 as in pyrite.
If you are dealing with stoney meteorites, they should contain olivine, pyroxene, and possibly plagioclase. Chondrites will contain small spherical bodies, resembling droplets, in a mineral matrix. The chrondules are distinctive and will consist of olivine, pyroxene, or both. Occasionally they display a barred texture. There are some distinct chemical signatures for meteoritic pyroxene, olivine, and feldspar, but that topic is too detailed to go into here. A good reference is Papike et al. (2003) Determination of planetary basalt parentage: A simple technique using the electron microprobe, American Mineralogist, 88, 469