Special birthdays

There are some birthdays that have special names in Japanese, although the significance is more often word play (or kanji play) than a significant milestone.

還暦 かんれき is 60 years old. 還 means 'to return' and 暦 means 'calendar'. The Chinese system of naming years after twelve animals is well known, but there is a lesser known cycle of ten 'heavenly stems'. The detail of this is for a different post, but the two systems combine together to create a 60 year cycle. Therefore, after 60 years you have lived through a full cycle, and the calendar returns back to the beginning again.

古希 こき is 70 years old. 古 means 'old', but not an old person, and here means 'long ago'. 希 means 'wish' usually, but it can mean 'rare' which it does here. The original way of writing the kanji with this meaning was 稀, but 稀 is now usually abbreviated to 希. The word comes from a poem: 
酒債は尋常行く処に有り 人生七十古来稀なり
Everywhere I go, a bar tab is common thing for there to be; a man who lives to 70 is a rare thing since long ago.
Basically, the man travels a lot and just as sure as he'll run up a debt in a pub, he'll not run into a man that's over 70.

喜寿 きじゅ is 77. 寿 means '(long) life'. 喜 means 'delight', and in Japan has an abbreviate form that looks like 七十七 (7, 10 and 7: 77) arranged as one character.

傘寿 さんじゅ is 80. 傘 umbrella may seem like a strange choice, but the character can be abbreviated to 仐, which is 八 (8) and 十 (10), hence 80.

米寿 べいじゅ is 88. Why 米, rice? Again it's the character's form. The top two dashes form an inverted 八, the cross is 十 and the bottom two strokes form a second 八. 88!

卒寿 is 90. そつじゅ It's a bit late for graduating, but 卒 (most commonly seen in 卒業) can be abbreviated to 卆, which is 九 (9) and 十 (10).

白寿 はくじゅ is 99. 白, white, isn't abbreviated to anything resembling kanji numbers. If 百 is 100 and 一 is 1, removing 1 from 100 leaves 99; removing 一 from 百 leaves 白. Therefore, 白 means 99. Clearly.

There are special names for other ages as well, but these are the most common ones. One thing to remember is that the ages are normally based on the old method of counting years: a new-born baby is one at birth, and you add a year at the new year. So you're actually 米寿 at 87.