Much of course depends on the companies involved.
But there's several common schemes in place:

• lease. Effectively like your lease car, you pay a fee per month or year over a contract period. At the end you either get a right to purchase the aircraft for a nominal price or it reverts to the lease company, or you extend the lease.
• purchase. Yes, you can pay cash. Especially older companies may have the cash reserves to pay for new aircraft out of pocket.
• bank loans. Either a bank or the manufacturer underwrites a loan with the aircraft as collateral. Pretty similar to a mortgage on your house
• trade in. Like buying a new car, you might get a fair price for that old aircraft you had sitting around
• subsidies. There's still a lot of national prestige involved in some countries with getting foreign airlines to fly your product. Some aircraft manufacturers as a result can deliver well under the list price to foreign airlines because such sales may be heavily subsidised by their governments.

Much of course depends on the companies involved, but there's several common schemes in place:

• Lease
  Effectively like your lease car, you pay a fee per month or year over a contract period. At the end you either get a right to purchase the aircraft for a nominal price or it reverts to the lease company, or you extend the lease.
• Purchase Outright
  Yes, you can pay cash. Especially older companies may have the cash reserves to pay for new aircraft out of pocket.
• Bank Loans
  Either a bank or the manufacturer underwrites a loan with the aircraft as collateral. Pretty similar to a mortgage on your house
• Trade in
  Like buying a new car, you might get a fair price for that old aircraft you had sitting around
• Subsidies
  There's still a lot of national prestige involved in some countries with getting foreign airlines to fly your product. Some aircraft manufacturers as a result can deliver well under the list price to foreign airlines because such sales may be heavily subsidised by their governments.