Raw sugar is from the first pressings of sugarcane and is coarser than white refined sugar. Think of raw sugar and its varieties--Demerara, Muscovado and Turbinado--as ingredients rather than just as a sweetening agent. They tend to add complexity to a drink and blend best with heavier, richer spirits. I especially like raw sugar syrups with rum and scotch.
Gomme syrup is made at a daunting percentage of sugar to water and uses gum Arabic, which acts as an emulsifier. Gomme syrup has an amazing velvety texture and can be used with classic, rich drinks like the Sazerac or Martinez. You can buy gum Arabic from here.
Grenadine stands in a class by itself. It's essentially simple syrup flavored with pomegranate, but it bears mentioning because it's called for in so many cocktails. A few great recipes exist online for making your own. Steer away from commercial brands that don't contain a drop of pomegranate, although there are some quality commercial mixers like Stirrings, which are quite good. Use grenadine in specified recipes and not as a general sweetener.
Orgeat syrup is a complete pain in the woo-hah to make but it has amazing results in classic Tiki drinks like the Mai Tai, among others. Flavored by almonds, flower water and sometimes apricot pits, Orgeat adds an exotic profile. Trader Vic's orgeat syrup is a somewhat reasonable substitute to homemade for those with limited time.
Honey and agave syrup actually work pretty well in the same instances, although by no means are they identical. Agave is from the agave plant and not surprisingly works great with mezcal and tequila. Honey gets a bit complicated when you consider different varietals like sourwood or buckwheat. For those who just reach for the tiny bear, try blending it in a two to one ratio with hot water to make it more pliable. Honey, citrus and gin work magically together.
Maple syrup and golden syrup (like a lighter molasses) are also great mixers with sour Bourbon drinks or Applejack. There's something in both of these sweeteners that blends particularly well with aged American spirits. There are different grades for maple syrup. I've found grade B makes a great drink but some prefer to step it up with grade A.
Along with maple syrup, there are other breakfast favorites--jams, jellies and fruit purees. The Omar Bradley calls for marmalade in place of sugar and fruit in an old fashioned and works great. Straight fruit purees often need additional sweetening. Jams, jellies and purees are typically one-offs in recipes.
Molasses and brown rice syrup both tend to make things taste like health food store cookies. If you don't know this flavor, it has an odd sweet/bitter taste--somewhere between sugar and shoes. Although either one might work in a random recipe here or there. The Black Stripe is a good example.