How can wine be not kosher?
It has to do with idolatry. In the past many nations used wine to worship their gods. Hashem wanted to separate the Jewish nation and wine away from the nations and their idolatry wine.
So, for a wine to be kosher, the production must be supervises by a Jew who is shomer Shabbos and must supervise that all the ingredients are kosher. (Including when wine needs to be kosher for Pesach). And it should have a hechsher.
According to the process of the wine, it is either cooked or not cooked. If the kosher wine is cooked, then a non-Jew may touch it or a Jew who is Mechalel Shabbat. And the wine is still fit to be drunk by Jews. But an uncooked Kosher wine that is handled/touched by a Gentile or a Mechalel Shabbat Jew is spilled and no longer able to be drunk by Jews. Cooked wine is called mevushal.
I know someone who can recommend some good Kosher wine. I believe Kosher moscato.
I have Manischewits wine and it's not that good but not as terrible as another kosher wine I had at a Shabbat dinner (I forgot the name of it). So, I believe there are worst things than Manischewitz. I think Kosher moscato is the way to go. Rashi is ok too