In some ways this is almost too simple to devote a post to, but I know that I resisted making syrup for a while, so my hope is that describing how easy it is will help you take the plunge.
Mint Simple Syrup is used in two very popular classic cocktails: the Mint Julep, and the Mojito. Further, though, it's a great way to vary numerous other cocktails, including making a simple highball with a nice flavor (e.g., bourbon, mint syrup, and club soda). I looked around for a while to try to buy mint simple syrup in a bottle, but couldn't find it. So, it's worth knowing how to quickly whip up a batch.
Aside from the actual creation of the syrup, I've learned a few tips. First, don't buy the mint until you're ready to make the syrup. It gets limp and sad fairly quickly, and starts turning black after a week in the refrigerator. Second, don't make the syrup until you're planning to make cocktails fairly soon. Even adding a capful of vodka only extends the lifespan to about two weeks, refrigerated. Third, unlike in the photo, use granulated sugar, not cubes. You can't really measure the cubes to get the right ratios.
So, when you go to the store, all you need to buy is granulated sugar and some fresh mint. By the way, people do this with brown sugar as well, but I've never tried that.
For your first batch, use one cup sugar and one cup filtered water. Put the water in a saucepan and bring to a boil. Reduce the heat to a simmer and stir in the sugar. Keep stirring until it dissolves completely. Next, strip the leaves off the mint sprigs and drop them in the syrup. For this quantity, I use just enough mint to cover the surface - for a larger batch, use proportionately more mint. Some recipes suggest chopping or otherwise doing violence to the mint. This is not necessary and it likely adds a bitterness you are not looking for. Instead, use a muddler or just a spoon to press the mint leaves against the bottom of the pan, rubbing them slightly. You don't need to do this very long, just once for each leaf. Then stir for another minute on the low heat.
Use a fork to pull out the mint leaves - give each forkload a chance to drain a little into the pan before discarding it. If you are planning on storing the syrup for any period of time, put a capful of vodka into the mixture as a preservative, and stir it in. Finally, let it cool, pour it into your storage container, and refrigerate.
If you didn't pay attention in Chemistry class, you might think that a cup of water and a cup of sugar would give you two cups of syrup, but that's not the way dissolving works. The product is about 1.3 cups. Some people like a richer ratio, like 2:1 sugar:water - you can experiment, but the only difference is whether you are watering your cocktail down a little bit. Most recipes use such small amounts of syrup that it probably doesn't matter.
Obviously, you could eliminate the mint and make your own plain ol' simple syrup, but I'd recommend against that, except perhaps for a party where you're going to use large quantities. The bottled stuff keeps vastly longer, which is important in the home bar.