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Blocks never stop generating?

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Hello. My understanding of Bitcoin is not perfect, so I would like to ask questions about its operation. When Generating, the user will gain 50 coins after creating 120 blocks. Why does the program continue generating blocks after that point? And why does one not gain another 50 coins after generating 240 blocks?

Also, I sent a few coins to someone to test the transaction system. Even though it was just a few coins, that transaction's status currently reads "166 blocks". It seems that it will continue incrementing indefinitely. Why is this?

I thank responders for increasing my knowledge.
The user gets 50 coins when he has created 1 block. The coins can be used (they "mature") when 120 blocks have been generated by the whole network after he created the block. The number of blocks of a transaction means the number of new blocks in the whole block chain after the transaction. The whole block chain grows ad infinitum.
So that transaction will eventually read "1000 blocks" or more? Is this information important?
Yes it will.

Importance is relative. You can use the number of blocks to calculate some statistics. Also, if you press ctrl+[numpad+], you will see the serial number and hash of the blocks. (At least on windows).
Where it says "# blocks" in the status column I'm changing it to say "# confirmations".  That might be clearer.

If you doubleclick on the transaction you get a little more information.