This is generally met with confusion by non-birders, including me. What it means is a bird's minimum age in years. If a bird is within the first year of its life, it's 1cy (unless it's an egg, or a chick if it hasn't fledged). If it's one or more years old, but you don't know how old, you should use 1cy+. If you know a bird is more than one year old but less than two years old, use 2cy. And on it goes. If that's a tad too obtuse, you can just choose from egg, chick, 1cy for fledged juveniles, and adult. (You can also leave Age blank.)
How would you even know how old a bird is? Some species have distinctive signs in their plumage in the first few years of life. For example, all blackbirds start out with brown plumage and dull-coloured beaks when they fledge in spring-summer. In autumn, the male birds moult their plumage to become black, except for their primary flight feathers on their wings which remain brown (have a careful look at the first photo above). At around this time, they'll take on the yellow-orange beak colour of adult birds. In the second year, male blackbirds moult again and take on their distinctive complete black plumage. So, it you'd got a brown blackbird with a dull beak colour, it's 1cy. If you see a black male with an orange beak in spring or summer that still has brown primary feathers, it's 2cy. If it's an all-black male, it must be more than a year old so it's 2cy+.
For older birds, your only hope of aging them is if some clever ornithologist caught it in the past and banded it. Whenever you see colour bands on a bird, it is exceptionally useful to record in your comments the colours and their order on both the right and left legs. If you find a dead bird with a metal leg band, it's also very helpful if you send in all the details including the band number to the National Banding Office at the Department of Conservation.
In these abbreviations, "cy" means "calendar year", which is a hang-over from northern hemisphere ornithology (and the Swedish web code origins of the current NZBRN system). Bird breeding seasons up north conveniently span one calendar year from spring to autumn, Seasons in the Southern Hemisphere do not behave so well with our Northern Hemisphere calendar. When you use NZBRN, "1cy" still applies to <1 year old birds, etc.