Melty Blood is the original game. The story supposedly follows Satsuki's route, which was never released with any of TM's games, which takes place one year after Tsukihime. The story and plot doesn't change with the subsequent expansion, but a lot of side stories are added (more Neco Arcs!). In the original Melty Blood, only six of 14 characters were available to be played, the other 8 had to be unlocked through playing Story Mode. This was later changed in an update for the game (aka the "Nero" patch), which unlocked all the characters from the start.
Sion is the main character of Melty Blood. She originally came to Shiki's town in order to find the cure for vampireism and defeat the Night of Walachia once and for all.
In Actress Again, she is responsible for the reenactment of Walachia.
Although this is supposedly Satsuki's route...
She never appears. She is only mentioned a total of three times in the full story, until ReACT. However, if you read into the game's story, her appearance could might as well been a fake created by Wallachia or White Len...
Melty Blood ReACT is an expansion for the original game (like Burning Crusade was to World of Warcraft). This expansion added 2 playable characters and 4 non-playable ones. I believe ReACT also adds an Arcade mode, which is a continuation of sorts of what happens in the Story Mode played out in fighting game style instead of visual novel style. Start with this game if you are just starting out with Melty Blood series and want the whole story (original Story Mode + Arcade Mode).
Melty Blood Act Cadenza is a further expansion of ReACT. This added 4 additional playable characters and various graphical and gameplay changes. Note that this version does NOT have a Story Mode.
Melty Blood Actress Again is a sequel to the previous game. It features many drasticly new mechanics, two new characters, and 3 different fighting styles for all known characters. Several secondary characters have had their storylines expanded upon to include what happened to them after the main storyline.