The real difference between all these kinds of cream have to do with their fat content. It breaks down like this (percents are averages):
Half-and-Half: 12% fat
Light Cream: 20% fat
Light Whipping Cream: 30% fat
Whipping Cream: 35% fat
Heavy Cream and Heavy Whipping Cream: 38% fat
As food scientist Harold McGee explains, "The proportion of fat determines both a cream's consistency and its versatility." As the fat content increases, the cream gets thicker and you can do more with it.
Creams with more fat will whip up better into a stable whipped cream, and they will also resist curdling when used to enrich soups. Creams with lower fat content are better used in beverages or for pouring over desserts.
If you can't find the specific cream needed for a recipe and are looking to substitute, it's always a safe bet to go for a cream with a higher fat content than the one called for. This ensures that everything on a molecular level will work out.
If a cream with higher fat not an option (or you're hoping to cut calories from a recipe!), try the cream with the fat content just below the one called for before experimenting with lighter creams.
And, hey, if you want to substitute whipping cream for half-and-half in your coffee, mum's the word.
Related: Tip: Stabilize Whipped Cream