If I understand correctly, these are a way to automate the execution of the financial part of normal contracts without the need for undue trust between parties. Some simple examples that might help (and I invite correction):
• I and a few investors agree to buy into an enterprise in several installments. A contract can be written to the blockchain that automates these payments. Should a majority of the investors decide to withdraw support, the payments will cease, but while more than 50% wish to continue support, everyone will continue.
• I have an enterprise that I am highly confident will generate profit. I can establish a smart contact that guarantees my investors a minimum ROI at specific intervals; should the balance fail to grow by the agreed margin, all investors will instead be refunded (in full or part) based on a scheme agreed a priori to be fair — say, a function of shares owned for how long.
• I and several other Ethereum miners agree to develop a scheme built on top of Ethereum for, say, a specific method of interoperability between the Ethereum blockchain and traditional futures markets. We have a specific idea for this, but agree we need to ensure that there is momentum in the project, and want those participating to have 'skin in the game'. We build a tontine such that all of us invest a non-trivial amount, and any investor becomes ineligible for payout if their hash power abandons the protocol. If this falls below a certain threshold, remaining investors can vote to continue or divide the pot and abandon the project.