Reds with lots of tannin (Nebbiolo, Aglianico, Sangiovese - Pinot noir excepted) and whites with lots of acidity (Chardonnay, dessert wines) can do so. Conversely, a younger wine will taste less vibrant than it should if kept for too long.

Only a few fine nectars benefit from long-term aging, most wines are actually at their best within a few years of release… but whether you wait 3 months or 10 years to enjoy a bottle, you’ll still need to follow 4 basic guidelines to store it properly. 

1.Ensure a consistent mild temperature 
10 to 13°C is the perfect one, not warm - heat being the Nº1 enemy that speeds up the maturation process - nor too cool neither - what can make corks harden while allowing air in. But even more critical is the steadiness: the good place is the one where the temperature is not likely to swing extremely or frequently; the expansion and contraction of the wine might push the cork out or cause seepage. 

2. Mind the humidity
We all should aim at 50 to 80% humidity since less damp would generate a risk of drying out for the corks and more could damage labels or cardboard boxes. If it’s not already the case naturally placing a pan of water in your storage area or a dehumidifier could fix conditions. 

3. Limit the direct light
Sunlight can make the wine heat up, degrade, and get old prematurely. Domestic light can also fade labels in the long run. 

4. Keep the bottle on its side (or not!)
Horizontal or vertical, that is the question. Traditionally, bottles have been stored on their sides; apart of being highly space-efficient, this position makes the (natural) cork stays in contact with the wine so it cannot dry out, cause oxidation and damage. But if the bottles have an alternative glass or a synthetic closure - or if the plan is to drink it « now-ish »! - there is definitely no reasonable or proven need for this. Upright, by the way, is the right choice for sparkling wines anyway. 

Now. You haven’t been blessed with the perfect basement that can double as a cellar? 
No problem. Just start with some simple racks in a safe place before upgrading your storage conditions together with your wine collection purchasing a small wine cooler or a wine cellar.