A few days ago, I had a question from someone who knows pistols, but not rifles "What exactly is CRF, and why do rifle snobs think it's so important" (I'm paraphrasing here).
Controlled Round Feed (CRF) is the property of certain weapons that keeps positive control of a round while the weapon is cycling, so that the round can be pulled back halfway through the loading cycle (at least once the round has been stripped from the magazine), or the weapon can be turned upside down without the round coming out etc...
The other common method is push feed, where the round is pushed ahead of the bolt. In a push feed gun, if you half stroke the bolt, then pull back, the round will be left where it was. If you turn a push feed gun upside down during the feed stroke, the round may fall out, or jam.
It is possible to double feed or otherwise jam a push feed gun with multiple cartridge cases or live rounds; it is nearly impossible to do so with a controlled round feed gun.
CRF guns are more difficult to single load rounds in, they are much more sensitive to magazines and feed angle, and much more sensitive to cartridge OAL.
An aside: push feed vs. controlled round feed applies to pistols as well. The 1911 with internal extractor is a CRF design, but many more recent auto pistol designs are push feed.
Critically, it is also much more difficult to design, and manufacture, a controlled round feed weapon than a push feed weapon; which is why the majority of rifles made over the last 40 years are push feed. It's also much of the reason why "rifle snobs" tend to give a disproportionately large weight on CRF in their overall evaluation of a rifle; not only for the direct properties, but also as an overall indicator that quality was valued over cost, in the design and manufacturing of the rifle.
Most tactical rifle users, and almost all dangerous game hunters, feel the positive control of the round, is worth the tradeoff of more sensitivity to external factors, and of more cost.
The Winchester model 70 classic, and pre '64 model 70, are both CRF; as is the Remington 40x (a special target/tactical version of the 700 action, used as the basis for benchrest and sniper rifles), and any true Mauser 98 actioned rifle (including most CZ rifles, Brno rifles etc...).
Winchester model 70 rifles made between 1964, and the reintroduction of the Model 70 "classic" in 1992 (non "classic" models are still push feed), as well as the Remington 700 (the most popular centerfire bolt action rifle in the world); are push feed.
Savage and Weatherby rifles are variable based on model and options. Most non magnum Savage and Weatherby are push feed, most magnum are CRF.
For any two given rifles, otherwise identical, you can expect that the CRF model will be about 20% more expensive.