Once again, Mr Irwin has the best info...
I would just like to add a couple more historical tidbits. One is that very influential gun writers of the day (especially Warren Page of Field & Stream) pitched the .24/6mm bore size to the shooting public as the ideal dual purpose round, tops for both varmint and deer. Page is given credit for being the principal originator of the .243, although his wildcat (.240 Page Pooper) was based on necking down military T-65 brass (later introduced by Winchester as the .308 Win) and used a 30 degree shoulder angle, not the 17 degree angle of the .308.
Remington, as noted, did consider the .244 to be primarily a varmint round, and made quality rifles with twists for that. After a couple years of slow sales, Remington changed the twist in their rifles to the 1 in 9 that would stabilize the 105gr Deer bullet, but by then the .244 had a reputation it never could shake off. A friend of mine had one of these later .244s, a Model 722, and it was an outstanding rifle. But the shooting public was convinced the .244 was no good for deer, whereas they were constantly told the .243 would do it all (and so it would).
A few years later Remington changed the name of the .244 to the 6mm Remington, and all 6mm Rem rifles have the fast twist needed for good performance with "deer" ammo. And then, Remington put their "new" 6mm in everything they made, all the models mentioned by Mike Irwin, and also the "budget gun" before the 788, the model 600 carbine.
Based on the 7x57mm Mauser case, the 6mm/.244 Remington has a slightly longer body, so it will hold a little more powder than the .243 Win, and a longer neck (.352 vs the .241 of the winchester case) and with careful handloading will yield about 200fps more velocity with the same bullets and barrel length.
As far as not finding the 6mm in Ruger's current line up, so what? Ruger has a long habit of taking the slow sellers off their production list for a few years at a time, then bringing tham back later, often in a "special" model. I'd be willing to bet that if you called Ruger and told them you wanted a certain rifle of theirs in 6mm, they would make it for you, even if they don't currently list that caliber as general production.
The 6mm Remington is far from dead, but it is also far from being the healthiest round on the market also. If you want the most performance from the 6mm bore size, without going to a Weatherby or a WSSSSSSSM or a complete wildcat, the 6mm is your best bet.