A very fair question and sadly some of the wine label and wine shop advice is ill-judged.
The easiest way to make a good stab at when to drink a wine is from history of previous vintages. Say for instance if a 1982 Ch. Palmer is just approaching it's peak, then for a comparable vintage (say) 2000 we could expect it to reach it's peak at a similar age (i.e. 24 years old). Thus you might see a drinking window of 2020-2030. This assumes similarity in vintage conditions, winemaking technique etc. In this scenario, we're in the realm of educated guesswork and many professional wine critics are quite good at this (albeit it's not too difficult to find a howler of a mistake in hindsight).
Without this history, tasting the wine can give plenty of clues, but the guesswork is much more variable. Although various elements of a wine are viewed as providing the elements needed for cellaring (acidity, tannins, good fruit) it's not that easy and it tends to come down to "is the wine in balance and of a style that warrants cellaring". This is very much a judgement call and I think even professional wine critics struggle to judge longevity of wines they've not encountered before.
Add to this the complication, that the "peak" of a wine is not a definite point in time as it varies between individuals and indeed between individual bottles (even from the same case, under the same conditions). Some people prefer primary traits of fruit, acidity, tannin with some supporting oak and revel in the pureness of the taste. Other (I'm more in this camp) prefer the wine to soften and take on secondary flavours from oak and the changes in the wine itself. Smells such as cedar, truffle, undergrowth or honey can appeal.
All this points to a very imprecise science.
What can you start looking for in wines you drink?
Tannins: If there's a dry astringency to the wine, sucking the moisture out of the mouth, then the tannins are still strong and would be expected to soften over time. Quite feasible to drink this sort of wine with a nice roast though. If the wine is very smooth and easy drinking, then there's probably not much tannin and is ready to drink.
Acidity: Very difficult to generalise, however I'll give it my best shot
Acidity level does not change over time. However over time it softens. If a wine is very sharp tasting, it's feasible this will soften over time, but only of benefit if the rest of the wine can support this cellaring period. A classic example to discover this is with riesling young vs. old. Another is Hunter Valley Semillon from Australia. The great ability of acidity is to deliver mouth-watering "freshness" to the wine as it ages. With insufficient acidity the wine becomes "flabby" and tends towards a dull, almost stale taste. Too much though, and it's still searing, even after the fruit has died.
Alcohol: I hear comments that alcohol (and by insinuation, the more the better) is required to enable a wine to age. I also hear comments that high alcohol wines don't age. I'm not sure I want to enter this debate, except to say that wines that are in balance seem to do ok. If a wine has a very spicy alcoholic nose and a burning sensation from the alcohol, then as the fruit dies away, there's a risk that the alcohol will dominate. If the fruit lasts, then there's every chance the alcohol will remain in balance.
What else can you do? Try some older wines. A good wine merchant might have a selection and it would be good to try an older version of a wine style you like. Try to gauge whether you prefer the younger wine or the older one, or like many people, you feel both have merits.
There really is an awful lot of guesswork in judging drinking windows and don't treat the critics views as gospel. They're very useful when you know they have a good understanding of the wine or at least the wine style. If you feel they're off subject then the credibility should drop.
For the most part I'm happy to go with a critics drinking window for a wine I have one bottle of and no prior experience. If I've tasted it once I might revise my opinion. If I've had 3-4 bottles over the last 6 years, I'll be quite a bit more confident of when I'd like to drink the remaining bottle(s).
I hope this is of use