DCR means Dynamic Contrast Ratio. http://en.wikipedia.org/wiki/Contrast_ratio
Contrast ratio is the difference between light and dark on a screen, expressed by a number.
If you take the brightest white on a screen, and the darkest black and compare the luminosity, you get the contrast ratio. For example, a 1000:1 contrast ratio means that the brightest white is 1000x brighter than the darkest black.
In a screen with a bad contrast ratio, lot of details would be lost in the dark areas.
So, is higher contrast ratio better? I believe yes.
The important thing to know about dynamic contrast, which isn't clearly mentioned in the wikipedia article, is that monitor manufacturers measure it by comparing an all-white screen with the backlight turned up to the maximum to an all-black screen with the backlight turned down to the minimum (or even off). With dynamic contrast turned on, the screen will auto-adjust the backlight based on what's showing, but all-white or all-black is so rare in actual usage that the dynamic contrast is basically just marketing hype. Higher static contrast (comparing white and black that are showing simultaneously) is good, but looking for higher dynamic contrast is pointless.
The Samsung S20B350H claims "Mega ∞ DCR" (in case it doesn't show up in my post, there's an infinity symbol in there that doesn't show up on Newegg), whereas the VH236H claims 20000:1 ASCR, where ASCR is short for "Asus Smart Contrast Ratio", which as far as I can tell is merely an alternate name for Dynamic Contrast Ratio. However, the Asus VH236H has higher native resolution than the Samsung S20B350H.
According to the monitor's specs, the contrast ratio is stated as "1000:1 / Mega DCR".
1000:1 refers to the static contrast ratio which is typical for monitors in general. "Mega DCR" is more or less just a marketing term to state very high dynamic contrast ratio.
Static contrast ratio is the true measure of contrast ratio for a monitor. DCR is more or less a marketing gimmick so huge numbers can be advertised. DCR more or less changes the brightness and contrast (within a narrow range) of the monitor based on what the monitor is displaying. I find it very annoying and I switch it off on my HDTV and my Asus VK246H monitor which has DCR (ASCR).