Example sentences
xiè谢xiè谢！bù不kè客qì气。
Thanks! You're welcome.
bù不kè客qì气，xiǎo小shì事。
No problem, it's nothing.
About this word
不 is "not" and 客气 means polite, ceremonious, standing on formality. The phrase is what you say when someone thanks you, and its literal sense is "don't be ceremonious", meaning none of that formality is needed between us. One sound rule matters: 不 is normally bù, but shifts to bú before a fourth tone, and 客 is fourth tone, so it comes out bú kèqi. Related uses: 别客气 for "don't mention it", and 您太客气了 when someone has given you something and you want to acknowledge the gesture.
The story behind the character
不
不 is the basic negator: put it in front of a verb or adjective and you have said no. 我不喝 I don't drink, 他不高 he isn't tall, 不客气 you're welcome. It covers the present, the future, and plain refusal. For the past, and for 有, Chinese switches to 没 (我没有钱). One sound rule: before a fourth tone, 不 is pronounced bú, as in 不是 bú shì.
客
客 is 宀 (a roof) over 各, which is there for the sound. Xu Shen defined it as 客, 寄也, someone lodging under your roof. 各 by itself once meant a foot arriving at a place, which is a lucky fit for a guest, but its job here is the sound. The visitor at your door is a roofline with a borrowed sound under it.
气
Three strokes of rising vapor. Xu Shen wrote 气, 云气也: "cloud vapor." Side note: the traditional 氣 is actually a different character (it meant "to gift grain"); they got merged when characters were simplified.