う・よう (Volitional)

よう (and う) are grammar constructions that function in a similar way to ましょう, or to be more precise, ましょう is the polite (conjugated) form of these auxiliary verbs. They are used as a casual way of expressing 'let's' when making suggestions, or 'I shall' when making statements.
よう is only one of the potential forms of this grammar point, and can actually be divided into two separate auxiliary verbs, う (used with う-Verbs), and よう (used with る-Verbs, する, and くる).
When attaching う to う-Verbs, simply replace the last kana with the equivalent お sound kana from within the same column (く becomes こ, う becomes お, す becomes そ, etc.), before adding う. For る-Verbs, replace る with よう.

• 今夜(こんや)は先輩(せんぱい)の試合(しあい)を見(み)に行(い)こう！
  Tonight, let's go watch the senpai's tournament!
• 早(はや)く食(た)べよう。
  Let's eat already!

As with many other verb conjugations, する and くる behave in a slightly different way to regular う-Verbs and る-Verbs. する will become しよう, while くる will become こよう.

• 私(わたし)が運転(うんてん)しようか？
  Shall I drive?
• 今日(きょう)はお金(かね)が無(な)いから明日(あした)こよう。
  I don't have money today, so let's come tomorrow.

The type of statements that う, and よう can be used for in Japanese are usually divided into three categories.
推量(すいりょう) - Expresses guesses (has a similar meaning to だろう, which itself is the よう form of だ)
意志(いし) - Expresses intentions (has a similar meaning to ～するつもりだ)
勧誘(かんゆう) - Expresses invitations or encouragement (has a similar meaning to 誘(さそ)う 'to invite')