IMPORTANT UPDATE: This article has been updated as of the prerelease of Magic 2014 on July 13; Please refer to the revisions below.
When one or more planeswalkers with the same subtype is in play, both will go to the graveyard when state-based actions are applied. How this works is different than the ‘legend rule’ as the name of the planesawlker does not matter. While there are still not a that many planeswalker cards, several planeswalkers have had more than one card created to represent them.
As of July 19th 2013, Magic 2014’s rules changes will alter how the Planeswalker Uniqueness Rule works. Starting on that day, the same player must control 2 or more Planeswalkers with the same subtype for the PUR to work; if I control a Chandra Ablaze while you control a Chandra Nalaar, they both exist on the field together with no issues. The second change is that if one player DOES control multiple Planeswalkers of the same subtype, they don’t all go to the graveyard; one lives. If I control a Chandra Nalaar, and I cast a Chandra Ablaze, I will choose which of the Chandras to ‘keep’ and which to ‘lose’. The remainder of the article is still correct.
This distinction from how legends work also means that you can have more than just four Chandras in your constructed deck; you can have four Chandra Nalaar and four Chandra Ablaze, both of which have the subtype Chandra. It also means that these two cards cannot both be ‘hit’ by something like Memoricide when a specific card name is chosen.