Ray Bradbury said it best when he described Dandelion Wine as "it tastes like spring." It's a little sweet, although you can make it dry if you want, and it has unidentifiable floral and grassy notes with a topnote of sunshine. And no, the flowers are not sweet enough by a long shot, they're actually considered a bitter herb.
To make one gallon, pick one gallon of wide-open blooming dandelion flowers, no stems. Put in a pot and add one gallon of boiling water, cover tightly, and let sit three days or so until the flowers float up to the surface and it smells like the underside of a mower deck. Strain out the solids and wring all the liquid out you can. Put over medium-high heat and add the juice of four oranges and four lemons. When it starts to simmer, slowly stir in 4 pounds of sugar for a sweet-ish wine, 3 for a dry wine (and note this recipe is for one gallon, the batch above started as 3.5 gallons of flowers, the juice of 14 lemons and 14 oranges, and 14 lbs of sugar). Cool to yeast-pitching temperature of below 80 degrees F, pour into a big bottle, add yeast and airlock, and forget about it completely for about a year. You can rack it off the lees in six months if you want, and certainly rack it before bottling or it'll have sediment in the bottles.
If you use a good white wine yeast like Red Star Montrachet (now renamed "classic cru" or some such nonsense) and a fruit wine yeast like Lalvin K-1111, you'll get more of the fruit character from the citrus juices, and thus more "sunshine" effect. Both of those yeasts top out around 13% ABV. You could try champagne yeast if you wanted to get to 16%, but you'd probably need to add more sugar to get there. This stuff isn't very good if it's too dry. It does need to ferment out completely, thus the year in the fermenter. It'll keep in corked bottles for years.
Finally, it's not a table wine. It is a digestive or aperitif, almost strong as port or sherry, so use aperitif glasses. The dandelions have a strong action on the liver and gall bladder that promotes good and complete digestion. Have a single glass of this after a too-heavy meal when you're feeling bloated, and in ten minutes you'll be ready to roll again. Mix it half and half with club soda or seltzer over ice and you have a lighter version that works even better. My wife calls it "The Recipe," as she's always had iffy digestion.
I made my first one-gallon batch of this from a recipe in Foxfire 2 around 22 years ago or so, and it helped her so much I've made at least a couple of gallons every year since. It's a pain to pick all those dandelions, but it's worth it. We seem to have overharvested the yard over the years, so where we used to get a gallon of flowers a day when the season started, we now only get a few cups per day and freeze them until we have enough for a big batch. I'm one of the few people in the world who wants more and bigger dandelions in his yard!