The Japanese form なくて is created by taking the negative ending ない, changing it to its adverbial form なく, and attaching the conjunction particle て. Just like the regular て form, it connects two clauses with an 'and' meaning, but it does so in the negative. なくて can be understood as 'is not (A), and...' or 'does not (A), and...', based on the type of word it follows.

• 時間(じかん)がなくて映画(えいが)を見(み)られなかった。
  I had no time, and couldn't watch the movie.
• この本(ほん)は面白(おもしろ)くはなくてつまらないです。
  This book is not interesting, and is quite boring. (は is often added between the stem of an adjective and ない, usually for emphasis)
• この駅(えき)は便利(べんり)ではなくて店(みせ)も少(すく)ないです。
  This station is not convenient, and there are few shops as well.
• かれは学生(がくせい)ではなくて先生(せんせい)でもありません。
  He is not a student, and he is not a teacher either.

It's important to remember that ない functions in two distinct ways. When attached to a verb, ない acts as an auxiliary verb. However, when used with nouns, な-Adjectives, or い-Adjectives, ない itself behaves like an い-Adjective (形容詞(けいようし)).

The main functional difference is that the い-Adjective version of ない can be preceded by the particle は for emphasis (becoming では with nouns and な-Adjectives). This is not grammatically possible when ない is functioning as an auxiliary verb after another verb.

• このパソコンは壊(こわ)れはなくて、いつもちゃんと動(うご)く。
  This computer is not broken, so it always works fine. (Grammatically incorrect, as は cannot be placed between a verb and ない)
• このパソコンは壊(こわ)れなくて、いつもちゃんと動(うご)く。
  This computer isn't broken, and it always works fine. (Natural Japanese)