There is single point to cross border between Georgia and Russia: Verchniy Lars .
It is open for the cars since March 1, 2010.
Note that you have to get the Georgian and Russian visas to cross the border.
You can't cross the Georgian border from South Ossetia or Abkhazia territories !
It is criminal in the Georgia , and you can get problems there.
Update:
According to this information , all you need to cross the border, is: If your country has an agreement with Georgia about no-visa crossing (such as Ukraine, Armenia, and so on), you simply must follow the rules of border-crossing of Russia and Georgia (see it on the official sites)
If your need visa to visit the Georgia, and you have it, and you follow the rules of crossing the borders - you will have no problems.
you can't get visa right at border in Verchniy Lars .
You can get it at all the others official borders of Georgia, but not there.
I think, this can be cause of problems for some peoples.
Also, there is a stats from various sites that there were some non-Russians and non-Georgians who cross the border.
travel.stackexchange.com/users/19/VMAtm
I have an updated answer to this question as of today the 12th of April 2012.
A Polish traveller at my hostel in Tbilisi tells me that foreigners from various countries are now being allowed to cross the Georgia/Russia border in both directions at Verchniy Lars.
Apparently a Russian law or rule or regulation was recently repealed and there had not been a restriction on this border crossing at the Georgian side.
I'll try to include a link if I can find one.
travel.stackexchange.com/users/140/hippietrail
I just met a friend yesterday who came back from a journey to Georgia.
He wanted also to cross the border to Russia but it wasn't possible.
They told him it is only allowed for Russian and Georgian citizens.
Don't know if they told him wrong facts, but afaik the border is closed.
travel.stackexchange.com/users/693/RoflcoptrException
It's illegal (although you CAN get from Georgia to Abkhazia and you CAN get from Abkhazia to Russia, but you'll almost certainly be arrested if you try to re-enter Georgia having entered from Abkhazia from Russia, although from the experience of friends going Georgia-Abkhazia-Georgia is doable and seemingly tacitly approved by both Georgian and Abkhaz authorities).
However, you CAN exit Georgia via Sarpi into Turkey (only a few hours' bus journey - maybe 4-5 IIRC) and then take the boat from Trabzon in Turkey to Sochi in Russia.
travel.stackexchange.com/users/2089/Fleur Flaneur
There is an account as of July 2016 from an Australian blogger, Nick of Pineapple Explorer , and his Ukranian friend crossing at Verkhniy Lars in a Georgian car, with photos to back it up.
It reportedly took them 12 hours to cross (waiting in line from 4am to 4pm) in high-season.
travel.stackexchange.com/users/10280/Jonathan Allard
I know the one border between the Republic of Georgia and Russia that is not in a rebel province was officially opened in the last couple of years... but reports are conflicting as to whether people not from ex CIS countries can cross.  So can somebody please confirm one way or another, is the border between Georgia and Russia open for travellers?
Is the border between Georgia and Russia open to citizens of other countries?
travel.stackexchange.com
<russia><georgia-country><caucasus><borders>
travel.stackexchange.com/questions/289
travel.stackexchange.com/users/140/hippietrail
There were restrictions in the past, but travellers from various countries are now being allowed to cross in both directions at Verchniy Lars. It has been considered illegal  to cross the border even though nowdays, visitors from various countries are  being allowed to cross the boarder from both sides. However, it has recently been said that there had never been a restriction on crossing the border from Georgia. 
Nowdays, forrigners from various counries are now being alowed t cross the boarder from both sides. However, it has recently ben said that there had never been a restriction on crossing the boarder from Georgia.  Crossing and visiting the border between Georgia and Russia is Illegal for travellers but it is possibile.