A comma (,) generally indicates pauses in speech. But, when it joins two clauses, it indicates a contrast between two ideas. In speech it is normal to draw attention to this contrast by a slight pause. A comma is the usual way of indicating this, although it is not obligatory:

• Sheila can eat anything and large quantities of it, but she never puts on weight.
  
• I'm going to make some New Year resolutions, but I don't suppose I'll keep them

A comma after but would be very unusual, Yong. Although we might pause after but if we are making up our minds about the consequences of something, this would not normally be indicated in writing, unless we want to indicate hesitation with a discourse marker such as er, reflecting informal speech:

• You can play in the fields down by the river, but not in the woods.
  
• You can play in the fields down by the river, but, er, not in the woods.

and as co-ordinating conjunction

Note that and when it is used as a coordinating conjunction indicates addition. In speech the addition of ideas is not normally marked by a pause, so no comma is needed, unless the items being connected are very long. Compare the following:

• The general announced that there would be a ceasefire and ordered all his troops to lay down their weapons.
  
• Jennifer decided to try the home-made steak-and-onion pie with mashed potato and buttered carrots, and David ordered the breaded fish with chips and peas.
  

defining relative clauses

Defining relative clauses identify the noun being referred to. They tell us which person or thing is meant. They follow immediately after the nouns that they modify with no pause in speech, so no comma is used in writing:

• The boy (that) I dated when I was at school has now married my best friend.
  
• The shares (that) he bought when he joined the company are now worthless.

Note that if we leave the identifying relative clauses out, the sentences do not have very much meaning:

• The boy has now married my best friend. (Which boy?)
  
• The shares are now worthless. (Which shares?)
  

Non-defining relative clauses do not identity what is being talked about. They simply add information about the person or thing already identified. Non-defining relative clauses are normally marked by pauses or intonation breaks and therefore also by commas:

• My friend Deborah, who was my closest friend in primary school, has now married my cousin.
  
• Trevor Nunn's production of 'Anything Goes', which opened at the National only two weeks ago, has nearly sold out.

Note that if we take these non-defining relative clauses out, the sentences still make perfect sense provided the context is known:

• My friend Deborah has now married my cousin.
  
• Trevor Nunn's production of 'Anything Goes' has nearly sold out.

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