In describing the rising and falling popularity of names, I've used the terms for the seasons to describe the four phases of a name's popularity fluctuation. (If you've spent any amount of time on my blog you know that Strauss and Howe do the same with the "seasons" of history, although their works are off-topic in this blog post.)
Spring - The name is increasing in popularity, and often falls into what is called on Nameberry the "hipster" phase when it is often well-liked by name enthusiasts and other similarly-minded people. These names may still seem a bit "out-there" for the general population.
Summer - The name is in its period of peak popularity and loses the "hipster" feel when it becomes most mainstream. In cases of really popular names, those not often around babies or young children may not realize the name's hotness but those that are do (this is what has inspired many of the titles in Pamela Redmond Satran and Linda Rosenkrantz's Beyond... series; in the late 1980s they published Beyond Jennifer and Jason, a decade later added "Madison and Montana" to the title, and now they have Beyond Ava and Aiden).
Autumn - The name is decreasing in popularity. Although these names are usually perfectly respectable to the general population, typically they are no longer considered "fashionable" and may sound dated. Names popular in the eras in which the current parents and grandparents were born that no longer are often in this season.
Winter - The name is steadily low in popularity. Names which were mainstream two seasons ago may sound old-fashioned to may people, but going for one of these names may actually put your child's name ahead of the curve if it "survives" the winter and doesn't become stuck in fashion limbo (think Bertha and Gertrude). Often names that don't fall as low are more likely to come back (for example Emma after being very popular circa 1880 never fell out of the top 500 and came back to near the top of the list in the 2000s, whereas the previous examples are all long out of the top 1,000).
A way you can determine when a particular name is or was in each season is to use a feature that graphs the name over time by the percentage of births (some good ones are The Baby Name Wizard's NameVoyager and Behind the Name's top popularity lists/graphs). Naturally, these discussions discuss the names using U.S. stats; internationally they may be different.
For example, if you were to show the aforementioned Emma, it's previous summer was in the 1880s-1890s, it had a long autumn lasting until the early 1960s, wintered for a little over 20 years until the mid-1980s when it started rising again into spring, and reached its current summer in the early 2000s.
For a boy's example, let's use one of my current favorites: Oliver. This one up until its winter followed a similar trajectory to Emma (although Oliver's peak was nowhere as high) with its height in the latter part of the 19th Century and then declining until the early 1960s. Oliver wintered longer though, not showing a major rising until the second half of the 1990s and still in its spring phase.
For a name on a different part of the cycle, I'll use an example of what Satran & Rosenkrantz call a "mom name": Amy. When the stats began in 1880, it was either late summer or early autumn being on its way down until the late 1920s or so, was at its low for about 20 years or so after, then had a 20-year or so spring, then its late 60s to mid-80s heyday, and then started to decline. In recent years the decline is leveling off a little bit; it's too soon to tell whether the name is now still in the autumn decline or now in winter again (the 2010 stats when they come out may help give a clue).