What is a Project Fork?
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A [*fork*](http://en.wikipedia.org/wiki/Fork_%28software_development%29) in software development refers to the event of an independent project spinning off from a software project. Such forks sometimes occur in the opensource sphere, when there are irreconcilable plans/goals within a project's community, then [often leading to a split in the community and two distinct projects thereafter](http://en.wikipedia.org/wiki/List_of_software_forks). In practice this takes form in the sourcecode being copied and henceforth being developed in a different direction independently by the forkers. For example in this conventional sense of fork, Litecoin is a fork of Bitcoin: Litecoin started as a copy of Bitcoin's code-base, but developed into an independent project (although still closely related).
Softfork and Hardfork in Bitcoin terminology
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The terms *softfork* and *hardfork* in Bitcoin describe compatibility breaking changes in the Bitcoin protocol: Should the community be irreconcilably divided about such an issue, the old version and the new version of Bitcoin could emerge as distinct projects thereafter. While both versions of the Bitcoin protocol are in use, the differences in acceptance may cause a lasting blockchain-fork, i.e. two distinct longest chains which are both considered valid by part of the network.
Softforks are [forward compatible](https://en.wikipedia.org/wiki/Forward_compatibility)
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**Softforks restrict block acceptance rules** in comparison to earlier versions.
![New valid blocks are a subset of old valid blocks][2]
The new rules allow a [subset][3] of the previous valid blocks, therefore all blocks considered valid by the newer version are also valid in the old version. If at least 51% of the mining power shifts to the new version, the system self-corrects:
Blocks created by old versions of Bitcoin Core that are invalid under the new paradigm might commence a short-term "old-only blockchain-fork", but eventually, they would be overtaken by the chain fork created under the new paradigm, as the hashing power working on the old paradigm would be smaller ("only old versions") than on the new paradigm ("accepted by all versions").
However, if less than 51% of the hashing power switches to the new version, it behaves like a hardfork, and the blockchain-fork will not mend, as the chain created under the old rules has more hashing power and is incompatible to the new rules.
Hardforks are not forward compatible
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![Hardforks extend set of valid blocks][4]
**Hardforks ease block acceptance rules** making previously invalid blocks valid in the new version. Obviously, this is not forward compatible as older versions will not accept the new blocks, causing the users of the old paradigm to remain on their own blockchain-fork indefinitely. To implement a hardfork, without a blockchain-fork, all users must switch to the new protocol consensually.
HF != Chain splits / chain forks
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Lately, I've seen "hardfork" being used interchangeably with "chain split". They refer to different things. **A chain split is the state of the network when mining is based on more than one chaintip at the same time.** This can happen for various reasons, including hardforks, softforks, buggy node software, and simply two miners discovering a block at the same height.
[1]: https://i.sstatic.net/H2w70.png
[2]: https://i.sstatic.net/axBEr.png
[3]: http://en.wikipedia.org/wiki/Subset
[4]: https://i.sstatic.net/v9Cme.png