The seven elements behind the days
Each Japanese day of the week is built from a kanji element plus 曜日 (youbi — day of the week). The elements are borrowed from classical Chinese cosmology: 日 (sun), 月 (moon), 火 (fire), 水 (water), 木 (wood), 金 (gold/metal), 土 (earth).
In order: 日曜日 Sunday (nichiyoubi), 月曜日 Monday (getsuyoubi), 火曜日 Tuesday (kayoubi), 水曜日 Wednesday (suiyoubi), 木曜日 Thursday (mokuyoubi), 金曜日 Friday (kin'youbi), 土曜日 Saturday (doyoubi).
A memory trick that actually works
If you already know the kanji for sun (日), moon (月), fire (火), water (水), tree/wood (木), gold (金), and earth/soil (土), you already know all seven days. There are no new characters to learn — just attach 曜日 to the element you want.
The abbreviated forms drop 曜日 entirely and just use the element character followed by 曜: 月曜 for Monday, 火曜 for Tuesday, and so on. Calendars and schedules often abbreviate further to just the single element: 月, 火, 水, 木, 金, 土, 日.
金曜日 (Friday) literally means "gold day." In modern Japanese culture, Friday has taken on the same end-of-week connotation as in English — プレミアムフライデー (Premium Friday) was a government initiative encouraging early finishes on the last Friday of each month.
Using days in conversation
To ask what day it is: 今日は何曜日ですか (kyou wa nan'youbi desu ka). To say "on Monday": 月曜日に (getsuyoubi ni). The pattern is simple and consistent across all seven days — learn one and you've learned the formula for all.