One of the fun things that you can do once you start getting into cocktails is making your own ingredients. Syrups like orgeat or lime cordial are pretty easy. A little up the difficulty scale are alcoholic infusions. The simplest are those like limoncello, where citrus peels are soaked in high proof alcohol, then diluted with sugar syrup. In the picture above you can see a couple of those in progress, with both lemon peels and grapefruit peels soaking in 95% alcohol. I usually buy Everclear because it's only $15 for a 750 mL bottle and it can be diluted down to whatever strength I want to use. While you can use 100-proof vodka or something like that in a pinch, the higher the proof of the alcohol, the more oils will be extracted from the peels. The same does not necessarily hold true for fruit infusions. The compounds that give fruits flavor are not necessarily as oily as the aromatic compounds in citrus peels. This means that different compounds will be extracted from the fruit depending on the alcohol concentration used to make the infusion. It's worthwhile to play around with the ABV of your infusion to see how this influences the final product. I usually let infusions sit for about a month, but you can make them go faster by giving the containers a swirl or a shake every so often to help the infusion proceed more thoroughly.
To finally get into some real chemistry, there's a good reason why agitating your infusion speeds up the process - as compounds are leeched from the peels, the concentration of those compounds in the area directly adjacent to the peels will build up, which makes for a smaller gradient between the solid and liquid phases. Moving things around spreads out the dissolved compounds, reestablishing a stronger gradient and causing the oils to extract more quickly.
After your infusion is done steeping, you need to filter the liquid away from the solids. A mesh strainer is a good first step. If you're using fruit, you may want to squeeze it to get out all of the delicious liquid, but you can also use it for making pie. To further clarify the liqueur, a metal or plastic coffee filter is a good way to get more of the fine particles. Actual coffee filters will also work, but they will also absorb some of the liqueur. Personally, I usually just accept that there's going to be some leftover solids in my infusions. It's not going to hurt you.
Once you're filtered your infusion, you have two options - either dilute the liquid with sugar syrup to make a liqueur or keep the infusion undiluted. The first is more traditional and makes for a handy ingredient that can be drunk straight or added to cocktails directly. As long as the resulting liqueur is over 20% ABV, it should be stable indefinitely without refrigeration. A high sugar content will also help to ward off any microbial growth. I personally tend to lean towards the second option, because I'm rarely interested in drinking liqueurs straight. By leaving the infusion in an un-diluted state, I can easily swap out the syrups that I use to sweeten the drink, with some interesting results. Alternatively, as in the cocktail at the bottom of this post, I can use the tinctures as base spirits instead of as liqueurs.
Infusions will often change in smell and flavor, even after the steeping is over and the liquid has been filtered. While they're technically ready to drink as soon as you've got them filtered, another couple of months will probably help to mellow and integrate the flavors and smells. Additionally, if you take the route of leaving the infusions undiluted, it's normal that you won't necessarily be able to smell a lot from the infusion. The high concentration of alcohol solubilizes the aromatic compounds very effectively. When the spirit is diluted with water, the aromatics become less soluble in the liquid and are then more volatile.
1 oz gin
1 oz mango tincture
0.5 oz lime juice
0.5 oz grapefruit juice
0.25 oz passionfruit syrup
0.25 oz simple syrup
1 dash Angostura bitters
The sip leads off with a bit of syrupy sweetness, which resolves into the tropical fruits of the passionfruit syrup and mango tincture. Near the end, the snappier flavors of gin, bitters, lime and grapefruit lead into the finish, keeping the drink from becoming insipid. I really like how the flavors flow past either other, blending briefly before passing off to the next set. The bitters, as so often they do, seem to be key to really amping everything up. Finally, for being a rather spiritous cocktail (remember, the tincture is equivalent to more than 2x the volume of normal 80-proof spirit), it ends up being rather balanced and refreshing.
Overall, this is a drink that I'm rather pleased with given that it was mostly an effort to use up some lime and grapefruit juice that I had sitting in the fridge.