What the term actually means
Speciality coffee generally refers to beans that are graded highly for quality, often from a single farm, region or cooperative. The focus is on careful growing, picking and processing, followed by roasting that brings out the bean's natural character rather than hiding it.
Because the coffee is traceable, you can often find out where it came from, what variety it is and how it was processed. That is part of why it tastes different from a generic supermarket blend.
What you might notice in the cup
Expect more variety. Beans from different countries can taste of chocolate and nuts, red fruit, citrus or caramel. Lighter roasts tend to show more acidity, which in coffee means a pleasant brightness, not sourness.
Freshness matters too. Good coffee is brewed soon after roasting and ground just before it is made, which gives a cleaner, sweeter finish with less harsh bitterness.
How to order if you are not sure
Tell the person making your drink what you usually like. If you enjoy milky, sweet drinks, a flat white or latte with a chocolatey, nutty coffee is a good start. If you like it black, try a filter or an americano and ask what is tasting good at the moment.
You do not need to know the jargon. Asking for something smooth, fruity or strong is a perfectly good way to be pointed towards a coffee you will like.
Making it at home
Buy whole beans in smaller amounts, store them in a cool, dark, airtight container and grind just before brewing. Fresh water and clean equipment make a bigger difference than most people expect.