Crying obsidian is a manufactured block, which means it doesn't generate naturally. It actually used to be in Minecraft itself, but was taken out. Its original use in the game was to reset the players' spawn point.[1]
It's texture was taken from the Minecraft wiki, but it seems to have an old obsidian texture, with water "drips" overlayed on to it. If an empty bucket is clicked on it, it will change to normal obsidian. When player is on fire, walking on this block does not put the flames out, but the player can right click the block to do so. This does not remove the block.
You can hold a water bucket, shift then right click on obsidian to change it to crying obsidian. You will then have an empty bucket.
You can then hold the empty bucket and right click crying obsidian to get a water bucket and change it back to obsidian.
It will not place a water source block down.