Rainy Season - Tsuyu 梅雨
Tsuyu (sometimes also called baiu) is what the rainy season is called in Japan, and for most of the country it happens from early June to mid-July - or from early May to mid-June in the south-westernmost prefecture of Okinawa.
Tsuyu is characterized, of course, by greater than normal rainfall, or at least very changeable weather, making always carrying an umbrella advisable even if it doesn't look like rain when you're setting out.
Tsuyu is a typical annual weather phenomenon in Japan but, being weather, it is unpredictable, hardly happening at all in certain years, and exceptionally wet in others.
In early spring, two bodies of warmer air, one from the North Pacific and one from South-East Asia, meet two bodies of cooler air, one over China and one over the Sea of Okhotsk. The this north-south collision between air masses of different temperature and humidity causes precipitation along the front where they meet. The southern air masses prevail, pushing the front gradually northward.
The onset of the rainy season (tsuyuiri or nyubai) is considered the end of spring and beginning of summer, and the end of the rainy season (tsuyuake or shutsubai) signals the start of full-on summer.
The two kanji for tsuyu/baiyu literally mean "plum" and "rain" and, of the several theories about the origins of the word, the most straightforward one is that the rainy season happens about the time when the fruit of the plum tree is ripening.
Another is that the kanji for bai was originally one that meant "mold" (黴), making for the phrase "mold rain," which refers to the moldiness that readily blights homes during this most humid of seasons, and that it eventually got changed to the less distasteful and more easily written kanji for "plum."
Rainy Season by Region
In a typical year, tsuyu happens in the various regions of Japan at the following approximate starting and finishing dates, from southwest to northeast Japan.
Okinawa: May 9 to June 23
Shikoku: June 5 - July 18
Chugoku Region (Okayama, Shimane, Hiroshima, Tottori, Yamaguchi prefectures): June 7 to July 21
Kinki Region (Hyogo, Kyoto, Mie, Nara, Osaka, Shiga and Wakayama prefectures): June 7 to July 21
Kanto: June 8 to July 21
Travel During the Rainy Season
Weather is a personal preference, but seeing Japan in largely cloudy, wet conditions is not most people's idea of an enjoyable sightseeing experience. As noted above, however, lots of rain and cloud are by no means guaranteed - just more likely. Also, starting in June, the rainy season is just after the annual peak travel period of Golden Week, in a trough of travel-related demand, so transport and entertainment facilities are operating at normal capacity, and accommodation and transport prices (e.g., for rental cars) are at their lowest.
One sight you may enjoy is the rainchains that hang from the eaves of many residences, shrines, and temples, with rainwater cascading down them, turning drear wetness into something eyecatching, beautiful and memorable.