To create the negative form of う-Verbs, you follow the familiar pattern of altering the final kana before attaching an ending. For negation, the ending you attach will either be ない for casual speech, or ません for polite speech.

• 先生(せんせい)は英語(えいご)で話(はな)さない。
  The teacher doesn't speak in English.
• 先生(せんせい)は英語(えいご)で話(はな)しません。
  The teacher doesn't speak in English.

Notice that the way the final kana changes depends on which negative ending you use. When adding ない, the verb stem's final kana shifts from a う-vowel sound to an あ-vowel sound in the same hiragana column. For ません, it shifts to an い-vowel sound instead. There is one important exception to this rule: verbs that end with the kana う itself. For the ない form, う changes to わ, not あ.

• 彼(かれ)はその本(ほん)を買(か)あない。
  He doesn't buy that book. (Incorrect conjugation using あ)
• 彼(かれ)はその本(ほん)を買(か)わない。
  He doesn't buy that book. (Correct conjugation using わ)

Although ない and ません are typically translated as 'do not', their meaning can shift based on context. They can express a general negative ('don't'), a refusal or decision ('won't'), or an inability to do something ('can't').

• ピアノは弾(ひ)きません。
  I can't play the piano. (Unintentionally not)
• 今夜(こんや)はまだ帰(かえ)らない。
  I won't go home yet tonight. (Intentionally not)

Keep in mind that ない and ません function as auxiliary verbs. Since ません is the negative form of ます, this means both endings can be further conjugated according to their own rules.