Several services make it easy to accept bitcoin payments, or a programmer can help you set up your own node. The most convenient way to accept bitcoin payments as a merchant is to use the services made available by exchanges like Coinbase and Bitpay, who make it simple enough to add a button to your website and to accept payments in person via NFC and QR codes. These exchanges have established what is called Full Nodes on the blockchain, which are slightly more efficient than using regular client software on the blockchain, and have optimized them for merchant services.
Similar to other bitcoin transactions using bitcoin wallets, each transaction will have it’s own address and funds are released by the payor using their personal key address, pushing the payment over to the payee after being validated by the blockchain. If you are particularly tech savvy or have a programmer that can help you get through it, merchants can establish their own Full Node.
It is relatively easy for anyone to allow their computer to act as a Full Node on the blockchain, which is different than mining and does not carry the same sort of monetary incentive. The community encourages new users to set up Full Nodes as much as possible since they help distribute work throughout the blockchain. This is done with open-source code that does not cost the person setting up the node anything but time (and electricity).
The benefits to having your own Full Node as a merchant are that your entire transaction history is not accessible to the company offering you merchant services and that you are also not at their mercy if their service goes down or is hacked. Bitcore is one example of open-source code that can help a merchant establish their own full node and wallet. It could also make sense for accounting purposes to convert bitcoin transactions into gift cards and use your existing accounting methods used for gift cards in your store, which is what Amazon is currently doing.