Some birders - many from the school of the great Peter Grant - are accustomed to using seasons to describe a bird's age: 1st winter, 1st summer, 2nd winter and so on and so forth. So a bird that was labeled as 1st winter yesterday (December 31st) is still a 1st winter gull today (January 1st). At some arbitrary point in the next few months, this bird will be referred to as "1st summer" (Note that the season a bird hatched in is not designated as its 1st summer. Hence, a bird in its 2nd calendar year will be labeled a 1st summer gull this upcoming June). The season system gets sticky in instances where transequatorial taxa are in question. For instance, a "1st winter" Sabine's Gull seen off the coast of California in November, and then found off the coast of Peru a few weeks later, will be in the company of resident species being referred to as "1st summer".
An increasingly popular method for "aging" birds - one that is based entirely on molt cycles and the plumages they produce - is the Humphrey and Parkes system (H-P; Auk 1959). Here, the focus is not on where or when a bird hatched, but rather, which feathers are being replaced and what happens thereafter in subsequent molts. Two broad molts are the basis for this system: prebasic and prealternate. Prebasic molts produce basic plumage, and prealternate molts produce alternate plumage. The H-P system avoids using "breeding" and "nonbreeding" to describe plumages as there are large inconsistencies in those terms. For example, are your local male mallards with green heads in breeding or nonbreeding plumage? Consequently, it's not very helpful to associate a bird's biology with its molts.
The H-P system was revised by Howell et al. a few years ago to promote congruencies across species based on events that take place in the 1st plumage cycle. There are four molt strategies that have been outlined in this revision (which is beyond the scope of this post). The results of this modification nicely sort most bird populations by molt strategy. One ambiguity with this system, however, is when the next prebasic molt commences. Let's consider a group of three gulls sitting together on a beach in Chicago during the breeding season. Suppose two of them hatched the previous year and one is a juvenile that just recently fledged. It's perfectly acceptable for one of the 2nd calendar year birds and the juvenile to be labeled 1st cycles, whereas the other 2nd calendar year bird may be labeled a 2nd cycle (simply because its next prebasic molt has commenced). Thus, it's not always obvious to a birder - without descriptions or photographs - which age cohort is being referred to when reading these things in print.
The H-P system is not designed to tell you how old a bird is. No system can without the use of a physical marker. Nor does the H-P system tell us whether a bird is in breeding or nonbreeding "plumage", but it does employ "aspect" as a way to describe the appearance of a plumage.
There are pros and cons to each of the systems above, but in my opinion, the H-P system - although somewhat didactic - is the closest thing we have to unfailing. A clarification note, Calendar Year is denoted by "cy", and this does not equate to "cycle".
Thanks to Chris Corben for some helpful comments on this post.