The reason you see black in the background of your image is because your image is not using the alpha channel.
Press N to open the panel on the image/UV editor and enable the Use Alpha option for the image:
Other things could that could be happening:
1.-The Alpha channel was not enabled during the import process.
If you are using the images as planes addon to import images that have an alpha channel you have to enable the Use Alpha option, and then choose the shader you need.
In 2.8 check the box for Use Alpha
For older versions of blender the options are:
In newer versions of blender the Use Alpha box might not be there but you can still select to have transparency as you import the image:
2.-Your Image has no alpha channel.
Open your file in an image editing software (gimp, photoshop, etc), delete the black background to make it transparent and save it again in a format that supports an alpha channel (JPG files do not support alpha, PNG, TIFF, OpenEXR do).
How can you tell if your image has an alpha channel?
At the bottom of the UV/Image editor's window you can select which channels to display:
Switching from RGBA to Alpha will allow you to see the alpha channel only
The white areas on the alpha will be opaque, the black areas will become transparent.
Just for comparison here's what the same image with no alpha channel looks like (As you can see it does not contain the outline of the leaf, just solid white):
AS AN ALTERNATIVE
If you don't want to edit your image, you can use the black information to control the transparency using a node setup like this where the parts of the image that are darker than 0.1 will be used as the mix factor for transparency. (Note that I inverted the order for the mix shader sockets)
Alpha transparency for the texture will only be displayed in rendered and material view.