Wine glass shape affects wine in three ways: how much surface area is exposed to air, how the liquid is directed onto your palate and how aromas are concentrated toward your nose.
Surface area matters because wine aromas are volatile compounds that evaporate from the liquid. A wider bowl exposes more liquid to air, releasing more aroma. This is why a large-bowled glass makes a young red wine smell more open than the same wine in a tumbler.
The taper at the top of the glass concentrates those aromas. A glass that is widest in the middle and narrows at the rim directs the aroma compounds into a smaller space directly under your nose. A glass that is perfectly cylindrical loses aromas in all directions. Even a small taper makes a measurable difference.
The rim of the glass determines where wine lands on your palate. A thin rim allows the wine to flow cleanly onto your tongue. A thick, rolled rim spills wine across your whole mouth at once, disrupting the sequence in which you experience the wine: first the front of the tongue (sweetness), then the sides (acidity), then the back (bitterness and tannin).
Champagne flutes are designed to preserve bubbles and concentrate the toasty, biscuit aromas from lees ageing. The narrow opening slows the loss of CO2. Many sommeliers now prefer a wider white wine glass for aged Champagne, arguing the aromas are complex enough to benefit from more exposure.
One decent universal glass handles most wines well. The improvement going from a tumbler to a basic wine glass is far greater than the jump from a basic glass to an expensive one.