Took some looking up for this, but 黒糖 is also written 黒砂糖、which is raw, unrefined sugar. The following links show how they're connected in the Infoseek dictionary.

The 液　part just makes it liquid form.

I think this is referring to molasses. It is commonly available in North America, but not readily available in Japan, as far as I know. I think people primarily use it in baking.

http://ja.wikipedia.org/wiki/糖蜜

--- Molasses (Japanese: 糖蜜) is a viscous "byproduct" of the processing of sugar cane or sugar beets into sugar.

--- The accurate English definition of 黒砂糖 is "muscovado" (although it is casually known as "brown sugar).

http://ja.wikipedia.org/wiki/黒砂糖
黒砂糖（くろざとう）とは、サトウキビの絞り汁を煮詰めて作る黒褐色の砂糖である。甘味料として用いる。別名に黒糖（こくとう）、マスコヴァド (英: muscovado) や、産地の奄美大島から称される大島糖（おおしまとう）などがある。
黒砂糖をbrown sugarと訳している和英辞典もあるが、これは多少糖蜜を残して精製した三温糖や赤砂糖、主にコーヒー用に用いる結晶状のコーヒーシュガーなどのことである。

http://en.wikipedia.org/wiki/Muscovado
Muscovado
Muscovado is a type of unrefined brown sugar with a strong molasses flavor.

---Here is a website for “Mayfair – the ingredients experts” featuring many types of sugars some of which are muscovado syrups:
http://www.mayfairfoods.com/sugars.htm

--- Muscovado syrup has a special flavor and used as an ingredient in cooking, baking, mixing drinks, etc. Here it talks about using it for mixing drinks:
http://www.adashofbitters.com/category/ingredients/syrups/
To my palate, the Muscovado syrup seems a little strong for most ingredients. It works best with a punchy whiskey, like a bonded rye, so I’ve been saving it primarily for Old Fashioneds and other whiskey cocktails. It occurs to me, though, that I haven’t played with it in a daiquiri. With the right rum, that might be worth trying.