There are some payment processors such as BIPS, BitPagos, BTCMerch, and a few more like them.
In detail, what do I require to get a website like that going?
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Sign up to join this communityCan you clarify whether you just need the ability to handle bitcoin-only transactions, or does your service need to be able to trade fiat currencies such as USD or EURO for bitcoins? If so, then you will need to invest in lawyers and find yourself a bank that is willing to extend you a loan, or supply your own funds to get it up and running.
If you're only needing to deal with bitcoins transactions, then I would suggest you learn about how blocks and transactions work at a low level. This is a good site for learning the nitty-gritty of how the blockchain stores data
http://codesuppository.blogspot.com/2014/01/how-to-parse-bitcoin-blockchain.html
Beyond that, understand how to set up a bitcoind server, and send JSON-RPC commands to it using the API explain here
https://en.bitcoin.it/wiki/Original_Bitcoin_client/API_calls_list
Once you know how these commands work, you'll be able to build software that will accept payments. Again, if you need to get fiat currencies involved that's you'll have to deal with legal/regulatory concerns, but it is possible.
You can use open source Bitcoin Payment Processor Libriary -
The easiest thing you could do, is just consider running an instance of bitcoind:
https://en.bitcoin.it/wiki/Bitcoind
With this, you can programmatically control the creation of wallets, addresses, and send/verify transactions ... the basis of a service such as bitpay. If you want to get a bit closer to the metal, you could look at using something like libbitcoin (which is what the guys at darkwallet are using):
http://libbitcoin.dyne.org/
Of course, you will have to know what you're doing when it comes to computer security, as running these on an unsecured server could result in a hacker stealing all of your funds.