They're popular for a few reasons. Its the cheapest military rifle you can buy, ammo is still cheap, they're everywhere and they have history behind them. 91/30 models are the most common, followed by carbines such as M38 & M44. Other models like M91 and 91/59 are out there but fetch a higher price and are less common. Some variations and features are more desirable to collectors but unless you have extensive knowledge of them its hard to know the differences. One feature that's easy to spot is an [SA] stamp usually found somewhere on the side of the receiver. The stamp signifies capture by Finland during their war with Russia and the Fins usually re-worked the rifles to squeeze out better accuracy.
Any Mosin variant with a bore and crown in good condition should be able to hit a man sized target at 100 meters and were usually sighted for 300 hence the reason some shoot high at the typical 100 yard range here in the US. Some people buy them expecting great accuracy but its really a crap shoot on getting one that will group tight with surplus ammo and no tweaking.
As far as which model is the best, that's tough to answer since collectors often have a preference for a certain arsenal, model, etc. but if you're looking for one to start with, just about any 91/30 from the recent batch of imports should be a good start. If you want more info, check the Mil Surp Collectors section. There's a sticky with lots of information and links.
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