なんか is a conversational expression that is closely linked to the adverbial particle など. In many situations, their meanings are interchangeable, and なんか is sometimes classified as an adverbial particle itself. なんて is a related expression, often thought of as a shortened form of phrases like などという, which means 'to say something like'.

Both なんか and なんて function similarly to など, translating to phrases like 'such as' or 'things like'. The main distinction lies in their usage with different word types. なんて is far more likely to appear after 用言(ようげん) (words that conjugate), which includes verbs, い-Adjectives, and な-Adjectives. Conversely, なんか is predominantly used after 体言(たいげん) (words that do not conjugate), which essentially means nouns.

• あんなに辛(から)いものを食(た)べるなんて、信(しん)じられないよ。
  To do something like eat food that spicy... I can't believe it.
• この映画(えいが)がこんなに悲(かな)しいなんて知(し)らなかった。
  I had no idea this movie was so sad.
• 冗談(じょうだん)なんて言(い)っている時間(じかん)はないよ。
  We don't have time for things like jokes.
• 宿題(しゅくだい)なんかあとでやればいいよ。
  Stuff like homework can be done later.

Despite the general tendency regarding 体言(たいげん) and 用言(ようげん), most native speakers would not consider it wrong to interchange なんか and なんて with any class of word.

• 僕(ぼく)なんか料理(りょうり)は全然(ぜんぜん)だめですよ。妻(つま)の方(ほう)がずっと上手(じょうず)です。
  When it comes to me, my cooking is terrible. My wife is much better at it. (Humility)
• あいつなんて絶対(ぜったい)に約束(やくそく)を守(まも)らないよ。
  A guy like him will never keep his promise. (Insult)

You should not let this nuance prevent you from using なんか or なんて, as in modern spoken Japanese, they are often used simply as filler words to bridge pauses in conversation.

• きのうの夜(よる)、なんか、あまり眠(ねむ)れなかったんだ。
  Last night, like, I just couldn't really sleep well.