Vodka is a grain alcohol (but can also be distilled from fruit/potatoes). In its pure form vodka is the most neutral spirit, but flavored vodkas are a popular category. Popular vodka cocktails: Martini, cosmopolitan, bloody mary
Gin is a grain alcohol similar to vodka but is flavored with herbs and botanicals, with Juniper berries being the most common flavor. Popular cocktails: Negroni, Tom Collins
Rum is distilled from sugar, which is what gives this spirit its sweet tones. Popular cocktails: Daiquiri, Pina Colada, Mojito, Mai Tai
Malt Whiskey (may also be spelled whisky) is made from malted grains (similar to beer except it is then distilled), and styles tend to vary based on the region it is produced. Irish whiskey is made in Ireland, Scotch whiskey in Scotland, bourbon in Bourbon County Kentucky, as well as Tennessee, Canadian, and Japanese whiskies. Popular whiskey cocktails: Manhatten, Old Fashioned
Tequila is distilled from the agave plant in Mexico. In 1978, the Tequila Regulatory Council made the regional distinction official. There are other similar agave spritis such as Mezcal. Popular cocktails: margarita, tequila sunrise
Brandy is distilled from fruit (usually grapes) in oak barrels. Cognac is a specific blend of brandy. Pisco and grappa are also common brandy styles.
Liqueurs tend to be the raw spirits above with added infusions or flavors.