My first choice would be "prickly pear cactus".

I've also seen "barbary fig tree" but it is not as common
http://www.websters-online-dictionary.org/definition/english...

http://www.libertystory.net/LSCONNECTMARKETS&CIV.htm

As for the leaves, they are called "pads". See here: http://www.desertusa.com/magoct97/oct_pa/du_prkpear.html


The link below has interesting info:

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Note added at 1 hr 6 mins (2005-02-03 04:32:29 GMT)
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I\'d specify \"cactus\" (prickly pear cactus) in order to differentiate it from the fruit (known as prickly pear).
http://www.wordreference.com/definition/prickly pear

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Note added at 1 hr 28 mins (2005-02-03 04:54:11 GMT)
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Here is another interesting link. It refers to it as \"nopal\" with the prickly pear and barbary fig as its fruits.

http://www.theworldwidegourmet.com/vegetables/south/nopal.ht...

http://www.gees.ac.uk/projects/outputs/fieldsim/flr_ppear.ht...

\"Tuna cactus\" here:
Opuntia ficus-indica has an annual life cycle. Prickly pear cactus (locally known in some parts as tuna cactus and barbary fig) is evergreen.
http://www.crescentbloom.com/Plants/Specimen/OO/Opuntia ficu...

DESCRIPTION: Opuntia is the largest group of Cacti with over 360 species ranging from Canada to Chile and Argentina. This group also includes several edible kinds. These plants are commonly known as Bunny Ears, Cholla, Prickly Pear, Barbary Fig, Tuna and Indian Fig. Opuntia is usually separated into two groups. The first group is Platyopuntia; those belonging to this group have round flattened joints, called pads. They are commonly known as Prickly Pears because they produce the spiny, usually edible, fruits. Some are grown for their edible pads called nopales or nopalitos. The spines of these pads must be singed off before they can be prepared for eating.
http://www.botanyworld.com/opuntia.html