You actually do use a keyboard. A plain 'ol English one, even..!
Japanese can be written with English lettering because most kana are just a consonant followed by a vowel.
For example, with "Sake", "さ" is "Sa" and "け" is "Ke", so the word "Sake" is "さけ".
If you wanted to type those two kana, you would just choose a Hiragana input method, and type "sake".
Kanji sounds are just combinations of Hiragana sounds, so you just type the Hiragana one after the other, and then use arrow keys to select the Kanji you want.
That's two of their four alphabets, their other two are "Romaji", which are the English letters themselves (Which is why a Japanese person would understand the keyboard), and Katakana, which has the same sounds as Hiragana, so you would just select that alphabet and type it like Hiragana.