A Caipirinha is one of the most refreshing drinks I know of. On a hot day, this one is right up there with mint juleps and gin and tonics. It is the national drink of Brazil and there is some work involved in the muddling, but it is well worth it.
Cachaca (at least a lot of it in Brazil) is a rough spirit and the best guess as to how this drink came about was Brazilians trying to find a way to make cachaca more palatable a couple hundred years ago. I think they did a fantastic job.
The half lime in this recipe is a Persian lime (the common lime in U.S. grocery stores). If you are using Mexican or Key limes use a full lime (or almost all of it).
I use turbinado (or Demerara) sugar instead of white sugar. The turbinado sugar gives the Caipirinha more body, complexity and depth. The turbinado crystals are quite substantial, and will abrade the lime skin nicely to release the lime oil. This oil combined with the sugar and lime juice give off an enchanting aroma. Give it a smell before the cachaca and ice go in.
Because of the size of the crystals, turbinado sugar takes awhile to dissolve. If you are making this drink to serve immediately just use granulated or superfine sugar. But if you can be patient I think turbindao sugar makes a superior Caipirinha. Don't try to muddle excessively to dissolve the sugar or you will release too much lime oil resulting in a sharp drink. Just muddle to crush the fruit and spend more time stirring.
If you need to make a lot of Caipirinhas you can pre-muddle a bunch of glasses to get a head start. These glasses can be left to sit to let the sugar dissolve passively for 5 or 10 minutes. I've seen this done in Brazilian bars where they have a dozen glasses premade with muddled limes and sugar. Just let these sit until you need them - then ice and cachaca and you're on your way.
A Caipirinha might taste a little concentrated at first, but as the ice melts a little, you will achieve an extremely refreshing beverage.
I use Pitu cachaca which is on the lower end of the price scale (at least in the U.S.). It works well. I might investigate a higher end bottling of cachaca such as Leblon to see if this improves the drink. Also, aged cachaca would be interesting to try.
If you make a Caipirinha with vodka instead of cachaca it is called a Caipiroska. If you use rum instead you have a Caipiríssima.