What happens when you swipe your credit card?

Note : The response is based on four party system for card processing here in US. Nevetheless, the card processing steps are generally the same in other parts of the world.

What happens when credit card is swiped?
Background
Credit cards are swiped at POS (point of sale). This scenario is also referred to as 'Card present'. On the other hand the online checkout scenarios are referred to as 'Card not present'.
Four party system

POS device
Hardware provided by acquiring bank to the merchant. POS devices support magnetic-stripe, EMV (Europay-Mastercard-VISA) chip, etc to read credit card details.
EMV
Credit card information is stored in a metallic chip implaned in a credit card. Traditonally this information also found in magnetic stripe.

Transaction flow in the four-party system

1. Purchase Initiation: The cardholder initiates a purchase by swiping or tapping at the merchant’s point of sale (POS) using their card card. As mentioned above, POS devices collects payment information from magnetic strip or EMV on the credit card.
2. Authorization Request: The merchant sends an authorization request (it contains card data, order/transaction amount, etc) to the acquirer, who forwards it to the card network (Visa or MC). In most cases a payment processor is involved in brokering the transaction requests, although it is not always the case.
3. Issuer Authorization: The card network routes the authorization request to the issuer, who checks the cardholder’s account for sufficient funds or credit limit and approves or declines the transaction. This step is essential to make sure the transaction amount requested doesn't exceed the available credit limit.
4. Authorization Response: The issuer sends the authorization response back through the card network to the acquirer, who relays it to the merchant.  
5. Transaction Completion: If the authorization request is approved, the transaction is completed, and the merchant provides the goods or services to the cardholder.
6. Clearing and Settlement: As a next step, the transaction details are submitted for clearing and settlement (either by the payment processor or the acquirer on the merchant side), where the issuer transfers funds to the acquirer, who then deposits the funds into the merchant’s account after deducting fees.