初夢

初夢 is the first dream seen after the New Year's. This could be the morning of January 1st or even as late as the morning of January 3rd.
The Kanji
First, understand the meaning, let's break down the kanji.
初
first time; beginning
Kun: はじ.め、 はじ.めて、 はつ、 うい-、 -そ.める
On: ショ
夢
dream; vision; illusion
Kun: ゆめ、 ゆめ.みる、 くら.い
On: ム、 ボウ
初 (with the hatsu reading) is used with many other words to indicate the "first time" of some significant event. Here are a few examples:
初耳 something heard for the first time [literally, first ear]
初恋 first love; puppy love
初詣 first shrine visit of the New Year
初舞台 debut; first stage appearance
初雪 first snowfall (of the season)
初霜 first frost of the year
初場所 New Year's sumo tournament in Tokyo
As you would expect, several 初 words deal with the first of the year.
The Three Dreams
Now, back to today's word. If your 初夢 involves one or more of three things, it is considered to be a good sign for the year. These three things are:

富士 Mt. Fuji

鷹 a hawk

茄子 eggplant
Mount Fuji, I get, but a hawk?! An eggplant?!
It seems no one really knows. But here are two theories:
- Mt. Fuji is Japan's most beloved and tallest mountain, the hawk is a clever bird, and nasu could also mean "to build up" or "to establish" or "to achieve": 成す
- Since Tokugawa Ieyasu was known to be fond of the three things, it may have been considered good luck during the Edo period to be like the Shogun.
The three are often grouped like this:
一富士、二鷹、三茄子
One, Mt. Fuji; two hawk; three eggplant
History of 初夢
The first recorded mention of 初夢 is from the 鎌倉時代 Kamakura period (1185–1333). While it was the first dream of Spring (and not January 1st), the idea is the same. Usage for the New Years began in the 江戸時代 Edo period (1603-1868).
Do you remember your first dream of the year?