I've spent some time reading around, and here is my guess as to what the site is describing:
Behind the scenes, achow101 (https://github.com/achow101) is running four Bitcoin clients:
* `Bitcoin Core` is running the client from https://bitcoin.org/en/ (https://github.com/bitcoin/bitcoin) (https://bitcoincore.org). This client does not support Segregated Witness, and it does not support larger block sizes (so blocks larger than 1 MB are ignored).
* `UASF (BIP 148)` is running the client from https://bitcoinuasf.org (https://github.com/UASF/bitcoin). This client supports Segregated Witness. This client does not support larger block sizes (so blocks larger than 1 MB are ignored).
* `btc1 (segwit2x)` is running the client from https://segwit2x.github.io (https://github.com/btc1/bitcoin). This client supports Segregated Witness, as well as changing the block size to 2 MB (blocks larger than 2 MB are ignored).
* `Bitcoin ABC (UAHF)` is running the client from https://www.bitcoinabc.org (https://github.com/Bitcoin-ABC/bitcoin-abc). It supports blocks of <8 MB in size (initially defaulting to 2 MB). This client does not support Segregated Witness.
As I understand things, right now there are two events which have hard-coded dates:
* At midnight UTC on August 1, the `UASF (BIP 148)` client will fork, in that it will start ignoring new mined blocks which do not support Segregated Witness.
* At 12:20 UTC on August 1, the `Bitcoin ABC (UAHF)` client might fork. I say might because the fork will be triggered by someone mining a larger-than-1-MB block (the "trigger block") on or after 12:20 UTC. (1 MB is the largest block that the `Bitcoin Core` and `UASF (BIP 148)` clients will recognize). At that point, `Bitcoin ABC (UAHF)` nodes will treat the "trigger block" as valid, and the fork has occurred.
Also, there is another event which may take place, depending on various conditions:
* If—sometime soon (before midnight UTC on August 1), and for a long-enough period—enough people indicate support, then the `btc1 (segwit2x)` node will fork with the `UASF (BIP 148)` node (in that both nodes will start supporting Segregated Witness). Sometime after that fork, the `btc1 (segwit2x)` node will fork off into its own fork, which will have support for 2 MB blocks.
And of course, there is one event which will happen at some point:
* `Bitcoin Core` will release a new version, adding support for Segregated Witness, or for a different block size, or both. `Bitcoin Core` nodes, when they upgrade, will then join one of the forks.
Anyway, I _think_ that's right, but this has gotten really complicated, and I'm trying to skip over the specific details, and the rationale (since, as tempay mentioned, that's been discussed recently).
If I misunderstood something, please let me know!