Do you know Tone Sandhi? 你知道变调吗？
-(Mandarin tone change rules) 中文变调规则
Once you have completed the basic tones of Mandarin, you will start to notice that they often change. For example, you will hear:
(Hello)你好【ní hǎo】 instead of 你好【nǐ + hǎo】；
(You are welcome )不客气【bú kè qì】in stead of【bù kè qì】
When and where they change? Actually, there are some specific rules that cause Mandarin tones to change in certain situations. These rules are called Tone Sandhi.
As a Chinese, I don't remember I learned these rules at school, I think that I unconsciously acquired it. But when I thought about it, I realized it was true and I even follow it.
Let’s look at this rules and I will add some examples for each of them:
* Rules for 3rd tone
1. A 3rd tone followed by another 3rd tone becomes a 2nd tone
When there are two third tones in a row, the first one changes to a second tone.
  →  
你好 (hello): nǐhǎo → ‘níhǎo’
很远 (very far): hěnyuǎn → ‘hényuǎn’
老虎 (tiger): lǎohǔ → ‘láohǔ’
2. A 3rd tone becomes a ‘half third tone’, if followed by a 1st tone, a 2nd tone or a 4th tone.
This ‘half third tone’ is a low-pitch tone that falls slightly.
 [1 /2 /4]→ [half third tone] [1 /2 /4]
考试 (exam): kǎoshì → kaoshì
草莓 (strawberry): cǎoméi → caoméi
卡车 (truck): kǎchē → kachē
* Rules for “一”
The character 一 (one) is normally first tone (yī), but this changes in two situations.
3. 一 is 2nd tone when followed by a 4th tone
[一]  → [一2] 
一个 ...(one …): yī gè → yí gè
一切 (all): yī qiè → yí qiè
4. 一 is 4th tone when followed by a 1st, 2nd or 3rd tone
[一] [1 /2 /3] → [一4] [1 /2 /3]
一般 (normally): yībān → yìbān
一起 (together): yīqǐ → yìqǐ
一杯水( A glass of water)yī bēi shuǐ → yì bēi shuǐ
*A rule for 不
不 is normally the 4th tone (bù), but there is one situation where this changes.
5. 不 is 2nd tone when followed by a 4th tone
[不]  → [不2] 
不是 (is not): bù shì → bú shì
好久不见（Long time no see）hǎo jiǔ bù jiàn→ hǎo jiǔ bú jiàn
不会 (will not, cannot): bù huì → bú huì
It is important for beginners to memorize these rules because sometimes textbooks will not remind you.