How can I send bitcoin from one address to another through the Bitcoin client/API?
I think you answered your own question in a way. That command will send from a specified account to an address, I'm sure you don't want to bother with accounts and just send bitcoins.

You can send from an address to another using a similar command.

sendtoaddress &lt;bitcoinaddress&gt; &lt;amount&gt; [comment] [comment-to]

Where &lt;bitcoinaddress&gt; is the address you want to send to and &lt;amount&gt; is amount of Bitcoins (up to 8 decimal places). The other two that are in brackets are optional and cosmetic.

You can issue these commands in a command line or using one of the many libraries for various languages. Check out the Bitcoin API reference.