The first aspect to understand concerning fortified wine is that it's, well, fortified. That is, spirit (grape brandy, often from the same locality in which the wine has been produced) has been added at some point in the processing regimen (usually during the fermentation process) to stabilize the wine. In the case of Port, this addition of spirit normally leaves varying degrees of residual (unfermented) grape sugar in the fortified wine.
Port destined to be bottled as Vintage Port usually comes from the very best lots of wine produced in very good to exceptional vintages. It accounts for a very small percentage of overal production, and normally spends a relatively short period of only 2 years in oak casks before being bottled. This procedure serves to retain great concentration of fruit, and renders such a wine a true candidate for long cellaring. Most Vintage Ports rarely even begin to open up before a decade to a decade and a half of time in the cellar, while the best wines from the greatest vintages can not only last, but improve, for many decades. Such examples count amongst the longest lived of all wines.
Bottles of Vintage Port should be treated like any fine wine, that is stored under proper cellar conditions in a stable position, the bottles resting on their sides so as to keep the corks moistened. You'll notice that bottles of Vintage Port invariably have little white paint brush marks near the bottoms of the bottles. These marks serve as indicators as to which side of the bottle has been orientated upward from the time of bottling.
When to open a bottle of Vintage Port? The key here is to understand the 4 key flavour/structural components that make up such a wine - the fruity characteristics derived from the grapes, the spirit that has been added to arrest the fermentation process and stabilize the wine, the tannins (some from the oak but, in the case of Vintage Ports, mostly from the grape skins themselves) and, last but not least, the underlying acidity. In a young Vintage Port, these components will have barely begun to integrate with one another, and will likely seem quite distinct on the palate. On the other hand, in the case of a mature Vintage Port, these components will have melded to a degree that makes them hard to distinguish from one another. Furthermore, the mature wine will have attained a proper degree of complexity and balance.
Vintage Port (unless your pockets are deep and you can afford to purchase an older vintage that's reached full maturity) requires patience! So why rush when you're finally ready to serve it? The most important 'trick' in serving a Vintage Port is to separate the wine from the accumulated sediment (the latter often considerable) in the bottle. There are fairly expensive Port decanting implements around (in which the bottle rests and is very slowly tipped by a hand crank mechanism) that work well. In lieu of them, I prefer to 'forecast' when I'm going to be serving a particular bottle, and try to stand that bottle upright a week (or at least a few days) before. This technique allows any loose sediment to slowly fall to the punt at the bottom of the bottle where, hopefully, it will remain during a gentle pouring stage. Transferring the wine into a decanter vessel for serving purposes also helps.
Most other sorts of Ports (Ruby, Late Bottled Vintage and Tawny) are ready to go from the time they're bottled, and will not improve with further ageing. Late Bottled Vintage Ports are nominally aged for 4 years or so in cask. Vintage Character and Crusted Ports, on the other hand, lie somewhere between those three and the Vintage Ports, in that they will throw sediment in the bottle. Single Vineyard Vintage Ports are often issued from small parcels of land known for their exceptional soil and exposure, and tend to be released in years in which the overall quality of the vintage doesn't quite warrant the release of Vintage Ports under a winery's primary banner. These can also be fairly long lived. Colheita Ports are somewhat analogous to Tawny Ports, except that they are produced from the harvest of a single year. Depending on how long a time they've spent in oak casks, they can vary quite substantially in style (the younger examples being close to Late Bottled Vintage Ports and the older examples being more like well aged Tawny Ports). As opposed to the cellaring of Vintage Ports, all of these types of Port (with the exception of Single Vineyard Vintage Ports) should be stored upright.
The longer that Port wines are aged in oak prior to being bottled, the more their essential fruity characteristics change from overt ripe berry fruitiness to a more subtle plum-like fruitiness. The wood maturation, with time, also imbues the wines with additional notes of vanilla, nuts and spices even as it softens the textural quality. Tawny Ports generally range in age from 10 to 40 years, with 20-year old examples often providing the optimum compromise between quality and price.
Sherry wines represent another entire universe, and are worthy of discussion elsewhere, as are the fortified (and much underappreciated) wines from the island of Madeira.