First of all, we know that the carbonate globules come from Mars, for two reasons. First, some of them are fractured, so that they had already formed when ALH84001 suffered some mighty shock. This shock must have been worse than the trauma of falling through the Earth's atmosphere and landing on the ground (which did not crack the meteorite), so it could not have happened while ALH84001 was on Earth. Second, the relative abundances of different isotopes of carbon and oxygen fixes the age of the carbonate in the range of a couple billion years, which much longer than it could have spent in space.
The polycyclic aromatic hydrocarbons, or PAH's, are unlikely to have come from anywhere but Mars for several reasons. First, they are found primarily in the interior of the rock -- if they had been picked up at any time in the 16 million years since it left Mars, they should be most heavily concentrated on the outer surface of the rock, but in fact this is precisely where they are least heavily concentrated. In addition, the particular mix of PAH's that occurs in ALH84001 is unlike that found anywhere else, including the industrial by-products in Earth's atmosphere. Finally, the concentration of PAH's found in the meteorite is about a thousand times higher than the highest concentrations found in the snows of the Greenland ice sheet, which is about the level one would expect to find in ALH84001 if they came from the atmosphere while it was on Earth.
The iron sulfide and magnetite particles interspersed in carbonate which seems to be partially dissolved present an interesting problem. To produce these particles chemically, you need very alkaline conditions -- but you need acidic conditions to dissolve the carbonate. Thus it seems unlikely that the origin of these particles is enirely chemical.
Some terrestrial bacteria, known as magnetosomes, are known to produce iron sulfide and magnetite particles, either inside themselves or (for some anaerobic bacteria) outside. The fossil remains of ancient magnetosomes, known as magnetofossils, are very similar in structure and composition to the particles of iron sulfide and magnetite found in the carbonate in ALH84001.
The ovoid objects found near the center of the carbonate globules resemble similar features found on some fossil rocks from southern Italy, which are generally accepted to be fossils of ancient "nanobacteria", the smallest known terrestrial bacteria. The tubular objects resemble fossils of filamentous bacteria, also found on Earth. The main difference is that the terrestrial fossils are larger, by about a factor of ten, than the structures on ALH84001.
Finally, the overall structure of the carbonate globules is similar to that of carbonate bundles produced by terrestrial bacteria both in the laboratory and in freshwater ponds. On Earth, the first carbonates to be produced (which therefore are near the center of the bundle) contain manganese; followed by iron carbonate, and finally a burst of iron sulfides (which would be found near the rim, which is the last part of the bundle to form). At this point, magnesium carbonate (also found in the rims of the carbonate globules in ALH84001) can be formed under the right conditions.
There are several possible alternative explanations of some of the peculiar features of ALH84001. Here are a some that the authors of the research article discuss.
Let's think first about the polycyclic aromatic hydrocarbons (PAH's). Although the mix and the level of concentration of PAH's does not match what would be expected from any other source, there is still the possibility that they came from contamination in the laboratory. But if so, there should be similar contamination of other meteorite samples which were processed in the same laboratory, and there has not been.
Next, look at the iron-sulfide particles. The quandary here is that there do not seem to be any possible conditions under which you could produce these particles and dissolve the carbonate rock -- the one requires alkaline conditions, the other acidic. You could imagine, though, that the conditions changed. Perhaps there was a period in which conditions were alkaline, when the iron sulfide particles were produced, followed by a period in which the envionment was acidic and the carbonate dissolved. Trouble is, the iron sulfide particles should also have been dissolved during the acid times, and there is no trace of this having happened.
The ovoid and tubular objects on the surface of the carbonate globules could conceivably have been produced by dissolution of the carbonate, but this has never been seen in terrestrial rocks, nor has dissolution ever produced these shapes in laboratory experiments.
One could imagine that these structures could have been produced while the rock was lying in the Antarctic ice, either by weathering or by Antarctic bacteria. However, the products of weathering in Antarctica are usually compounds that contain plenty of oxygen -- which ALH84001 does not -- and ALH84001 does not seem to be weathered anyway. As for Antarctic bacteria, the researchers tried culturing one of the chips, and nothing grew. This means that the chip is sterile, so if Antarctic bacteria had invaded it, they are gone now. But if bacteria had colonized the rock, they should still be there. In addition, if bacteria had gotten into ALH84001 while it was in Antarctica, then they should have gotten into other meteorites there also, leaving PAH's behind when they die. But most other meteorites from the area contain no PAH's.
There is also the possibility that the ovoid and tubular objects were somehow produced in the laboratory while they were being analyzed or prepared for analysis. But no such structures were found on other rocks which were run through the same procedures (including a chip of rock from the Moon).