There are lots of different visas available...
For this purpose, you could probably get away with a transit visa. It's generally valid for a day or two, but maybe longer if your means of entering/leaving the country is very slow (I can recommend the trans-siberian!). Otherwise (and for more flexibility) get a tourist visa.
Each Russian embassy seems to have a great deal of independence. Expect opening hours and fees (and often paperwork requirements) to be quite arbitrary, although US citizens are reliably charged more than others. Take every relevant piece of paper that you have.
I once found myself needing a visa urgently, without all the documents (inward and outward tickets, proof of accommodation) that that the consular clerk wanted. I said the magic words "can I pay extra...?" and had a visa (and entry and exit permit) within 15 minutes. Don't count on that kind of activity, though.
All visas must be "registered" on arrival; if you're staying in a hotel they will offer to do this for you, although certain other agencies can too.
You might or might not get fined if trying to leave with an unregistered visa, and the fine could well vary according to departure point. The same for other incorrect paperwork; entry/exit permits (they come with the visa) or immigration or customs forms (which are usually handed out before arrival).
One guy I travelled with managed to leave without a customs deklaratsia, simply explaining that an official had taken both copies on arrival; another guy left the country with rather more money than when arriving (he got carried away with an ATM). Both of these are supposedly offences, but both got away with little more than a raised eyebrow.
used to (and probably still does) have a special consular office offering special while-u-wait visas before going through passport control. However, the requirements are fairly awkward, and it's not really a good idea to rely on something like this. Any given airline might not let you board a flight to SVO
without a visa, anyway..?