What are they?

A Smart Contract is an arrangement that is documented and stored on the Blockchain which automatically undertakes certain procedures in accordance with the terms of that Smart Contract.
Unlike a standard legal contract (which says how certain parties must act and relies on those parties to comply with the contract), a Smart Contract is capable of enforcing the terms of the Smart Contract itself (through computer code) and undertakes certain procedures when specific conditions are met (e.g. when some event occurs in the real world or a specified value of a certain exchange good is triggered).
What's the fuss?

Standard vs Smart Contracts
Standard contracts may need to be monitored
Requiring efforts of monitoring to ensure compliance by the counter-party
Smart contracts require no or very limited monitoring
Being automated and self-sufficient
Standard contracts can be difficult to enforce
Facing difficulties in making those that signed the contract adhere to their contractual obligations (especially given that an element of trust is usually required)
Smart contracts automatically carry out certain actions
Reducing the need for any party to make another party act in a specific way
Standard contracts can be hard to keep track of
Having multiple document versions (e.g. draft, unsigned, signed and amended versions) can be difficult to keep track of, and may lead to fraudulent activity (e.g. forged signatures and signing false dates)
Smart contracts are self-verifiable, tamper-proof and accurate
Reducing the likelihood of future disputes, confusion or fraudulent activity.
An example

A simple example of where a smart contract could prove useful is to give someone the right to use an image (i.e. image licence agreement) as follows:
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1. Insert details associated with use of image
The owner of the image (i.e. Licensor) inputs details in the Smart Contract including payment procedure, ways the image can be used and how long for etc. (i.e. the Terms).
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2. Smart Contract is signed
The person who wants to use the image (i.e. Licensee) signs and agrees to the Terms of the Smart Contract.
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3. Smart Contract collects payment
The Smart Contract itself collects the fee that the Licensee has agreed to pay (when signing the Smart Contract) in order to use the image (often in the form of a cryptocurrency known as "Ether"), being the Licence Fee.
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4. Smart Contract grants right and image
The Smart Contract will facilitate the grant of rights and access to the image to the Licensee in accordance with the Terms of the Smart Contract.
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5. Smart Contract facilitates payment
The Smart Contract will transfer the Licence Fee to the Licensor in accordance with the Terms of the Smart Contract.